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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-sec 28201* Define the secant function. We define it this way for cmpt 4226, which requires the form  ( x  e.  A  |->  B ). The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  sec  =  ( x  e.  {
 y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1 
 /  ( cos `  x ) ) )
 
Definitiondf-csc 28202* Define the cosecant function. We define it this way for cmpt 4226, which requires the form  ( x  e.  A  |->  B ). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  csc  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1 
 /  ( sin `  x ) ) )
 
Definitiondf-cot 28203* Define the cotangent function. We define it this way for cmpt 4226, which requires the form  ( x  e.  A  |->  B ). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  cot  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x )  /  ( sin `  x ) ) )
 
Theoremsecval 28204 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  =  ( 1  /  ( cos `  A ) ) )
 
Theoremcscval 28205 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  =  ( 1  /  ( sin `  A ) ) )
 
Theoremcotval 28206 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  =  ( ( cos `  A )  /  ( sin `  A ) ) )
 
Theoremseccl 28207 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  CC )
 
Theoremcsccl 28208 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  CC )
 
Theoremcotcl 28209 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  CC )
 
Theoremreseccl 28210 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  RR )
 
Theoremrecsccl 28211 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  RR )
 
Theoremrecotcl 28212 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  RR )
 
Theoremrecsec 28213 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( cos `  A )  =  ( 1  /  ( sec `  A ) ) )
 
Theoremreccsc 28214 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( sin `  A )  =  ( 1  /  ( csc `  A ) ) )
 
Theoremreccot 28215 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( 1  /  ( cot `  A ) ) )
 
Theoremrectan 28216 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( cot `  A )  =  ( 1  /  ( tan `  A ) ) )
 
Theoremsec0 28217 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 28218 Prove the tangent squared secant squared identity  ( 1  +  ( ( tan A ) ^ 2 ) ) = ( ( sec  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 25-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( 1  +  (
 ( tan `  A ) ^ 2 ) )  =  ( ( sec `  A ) ^ 2
 ) )
 
Theoremcotsqcscsq 28219 Prove the tangent squared cosecant squared identity  ( 1  +  ( ( cot A ) ^ 2 ) ) = ( ( csc  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 27-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( 1  +  (
 ( cot `  A ) ^ 2 ) )  =  ( ( csc `  A ) ^ 2
 ) )
 
19.23.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 28220 If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3706 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
19.23.6  Not-member-of
 
TheoremAnelBC 28221 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using 
e/. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  A  =/=  B   &    |-  A  =/=  C   =>    |-  A  e/  { B ,  C }
 
19.23.7  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 28225 and df-dp2 28224 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10339.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 28222 Constant used for decimal fraction constructor. See df-dp2 28224.
 class _ A B
 
Syntaxcdp 28223 Decimal point operator. See df-dp 28225.
 class  period
 
Definitiondf-dp2 28224 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10339. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 28225* Define the  period (decimal point) operator. For example,  ( 1 period 5 )  =  ( 3  /  2 ), and  -u (; 3 2 period_ 7_ 1 8 )  = 
-u (;;;; 3 2 7 1 8  / ;;; 1 0 0 0 ) Unary minus, if applied, should normally be applied in front of the parentheses.

Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers.

The RHS is  RR, not  QQ; this should simplify some proofs. The LHS is  NN0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression  -u ( A period B ) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

 |-  period  =  ( x  e.  NN0 ,  y  e.  RR  |-> _ x y )
 
Theoremdp2cl 28226 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  -> _ A B  e.  RR )
 
Theoremdpval 28227 Define the value of the decimal point operator. See df-dp 28225. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
 
Theoremdpcl 28228 Prove that the closure of the decimal point is  RR as we have defined it. See df-dp 28225. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  e.  RR )
 
Theoremdpfrac1 28229 Prove a simple equivalence involving the decimal point. See df-dp 28225 and dpcl 28228. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  =  (; A B  /  10 ) )
 
19.23.8  Signum (sgn or sign) function
 
Syntaxcsgn 28230 Extend class notation to include the Signum function.
 class sgn
 
Definitiondf-sgn 28231 Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over  RR* (df-xr 9080) instead of  RR so that it can accept  +oo and  -oo. Note that df-psgn 27283 defines the sign of a permutation, which is different. Value shown in sgnval 28232. (Contributed by David A. Wheeler, 15-May-2015.)
 |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 , 
 0 ,  if ( x  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgnval 28232 Value of Signum function. Pronounced "signum" . See df-sgn 28231. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgn0 28233 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (sgn `  0 )  =  0
 
Theoremsgnp 28234 Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR*  /\  0  <  A ) 
 ->  (sgn `  A )  =  1 )
 
Theoremsgnrrp 28235 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
 |-  ( A  e.  RR+  ->  (sgn `  A )  =  1 )
 
Theoremsgn1 28236 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn `  1 )  =  1
 
Theoremsgnpnf 28237 Proof that the signum of  +oo is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 +oo )  =  1
 
Theoremsgnn 28238 Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR*  /\  A  <  0 ) 
 ->  (sgn `  A )  =  -u 1 )
 
Theoremsgnmnf 28239 Proof that the signum of  -oo is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 -oo )  =  -u 1
 
19.23.9  Ceiling function
 
Syntaxccei 28240 Extend class notation to include the ceiling function.
 class
 
Definitiondf-ceiling 28241 The ceiling function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4.

By convention metamath users tend to expand this construct directly, instead of using the definition. However, we want to make sure this is separately and formally defined. Proof ceicl 11187 shows that the ceiling function returns an integer when provided a real. Formalized by David A. Wheeler. (Contributed by David A. Wheeler, 19-May-2015.)

 |- =  ( x  e.  RR  |->  -u ( |_ `  -u x ) )
 
Theoremceilingval 28242 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A ) )
 
Theoremceilingcl 28243 Closure of the ceiling function; the real work is in ceicl 11187. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  e.  ZZ )
 
19.23.10  Logarithms generalized to arbitrary base using ` logb `
 
Theoremene0 28244  _e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  0
 
Theoremene1 28245  _e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  1
 
Theoremelogb 28246 Using  _e as the base is the same as  log. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  ( A  e.  ( CC  \  { 0 } )  ->  ( _elogb A )  =  ( log `  A ) )
 
19.23.11  Logarithm laws generalized to an arbitrary base - log_

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear.

This supports the notational form  ( (log_ `  B ) `  X
); that looks a little more like traditional notation, but is different from other 2-parameter functions. E.G.,  ( (log_ `  10 ) ` ;; 1 0 0 )  =  2

This form is less convenient to work with inside metamath as compared to the  ( Blogb X
) form defined separately.

 
Syntaxclog_ 28247 Extend class notation to include the logarithm generalized to an arbitrary base.
 class log_
 
Definitiondf-log_ 28248* Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as  ( (log_ `  B ) `  X ) for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.)
 |- log_  =  ( b  e.  ( CC  \  { 0 ,  1 } )  |->  ( x  e.  ( CC  \  { 0 } )  |->  ( ( log `  x )  /  ( log `  b
 ) ) ) )
 
19.23.12  Miscellaneous

Miscellaneous proofs.

 
Theorem5m4e1 28249 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 5  -  4 )  =  1
 
Theorem2p2ne5 28250 Prove that  2  +  2  =/=  5. In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase  2  +  2  =  5 has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 2  +  2 )  =/=  5
 
Theoremresolution 28251 Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.)
 |-  (
 ( ( ph  /\  ps )  \/  ( -.  ph  /\ 
 ch ) )  ->  ( ps  \/  ch )
 )
 
19.24  Mathbox for Alan Sare
 
19.24.1  Supplementary "adant" deductions
 
Theoremad4ant13 28252 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant14 28253 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant123 28254 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ta )  ->  th )
 
Theoremad4ant124 28255 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ta )  /\  ch )  ->  th )
 
Theoremad4ant134 28256 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ta )  /\  ps )  /\  ch )  ->  th )
 
Theoremad4ant23 28257 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant24 28258 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant234 28259 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ta 
 /\  ph )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant12 28260 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  ps )  /\  th )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant13 28261 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant14 28262 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant15 28263 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant23 28264 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant24 28265 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant25 28266 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant245 28267 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  et )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant234 28268 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  ps )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant235 28269 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  ps )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant123 28270 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ch )  /\  ta )  /\  et )  ->  th )
 
Theoremad5ant124 28271 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ta )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant125 28272 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ps )  /\  ta )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant134 28273 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  ps )  /\  ch )  /\  et )  ->  th )
 
Theoremad5ant135 28274 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  ps )  /\  et )  /\  ch )  ->  th )
 
Theoremad5ant145 28275 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( (
 ph  /\  ta )  /\  et )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant1345 28276 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ( ( ph  /\  et )  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theoremad5ant2345 28277 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ( ( et  /\  ph )  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
19.24.2  Supplementary unification deductions
 
Theorembiimp 28278 Importation inference similar to imp 419, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ch )
 )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theorembi2imp 28279 Importation inference similar to imp 419, except the both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theorembi3impb 28280 Similar to 3impb 1149 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  <->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi3impa 28281 Similar to 3impa 1148 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  <->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi23impib 28282 3impib 1151 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ( ps 
 /\  ch )  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13impib 28283 3impib 1151 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ( ps  /\  ch )  ->  th )
 )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123impib 28284 3impib 1151 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ( ps  /\  ch )  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13impia 28285 3impia 1150 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ps )  <->  ( ch  ->  th )
 )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123impia 28286 3impia 1150 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ps )  <->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi33imp12 28287 3imp 1147 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi23imp13 28288 3imp 1147 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13imp23 28289 3imp 1147 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi13imp2 28290 Similar to 3imp 1147 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ( ch 
 <-> 
 th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi12imp3 28291 Similar to 3imp 1147 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi23imp1 28292 Similar to 3imp 1147 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123imp0 28293 Similar to 3imp 1147 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
19.24.3  Conventional Metamath proofs, some derived from VD proofs
 
Theoremiidn3 28294 idn3 28425 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ch )
 ) )
 
Theoremee222 28295 e222 28446 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ( ps  ->  ta ) )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoremee3bir 28296 Right-biconditional form of e3 28558 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ta  <->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
Theoremee13 28297 e13 28569 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   &    |-  ( ps  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ch  ->  ( th  ->  et ) ) )
 
Theoremee121 28298 e121 28466 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ta )   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ch  ->  et ) )
 
Theoremee122 28299 e122 28463 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ( ch  ->  ta ) )   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ch  ->  et ) )
 
Theoremee333 28300 e333 28554 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )   &    |-  ( th  ->  ( ta  ->  ( et  ->  ze ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ze )
 ) )
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