P. 365 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pm13.195 36401*	Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3330. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
|- ( E. y ( y = A /\ ph ) <-> [. A / y ]. ph )
Theorem	pm13.196a 36402*	Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( -. ph <-> A. y ( [ y / x ] ph -> y =/= x ) )
Theorem	2sbc6g 36403*	Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( ( A e. C /\ B e. D ) -> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) )
Theorem	2sbc5g 36404*	Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( ( A e. C /\ B e. D ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> [. A / z ]. [. B / w ]. ph ) )
Theorem	iotain 36405	Equivalence between two different forms of iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
|- ( E! x ph -> |^| { x | ph } = ( iota x ph ) )
Theorem	iotaexeu 36406	The iota class exists. This theorem does not require ax-nul 4556 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( E! x ph -> ( iota x ph ) e. _V )
Theorem	iotasbc 36407*	Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define iota in terms of a function of ( iota x ph ). Their definition differs in that a function of ( iota x ph ) evaluates to "false" when there isn't a single x that satisfies ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( E! x ph -> ( [. ( iota x ph ) / y ]. ps <-> E. y ( A. x ( ph <-> x = y ) /\ ps ) ) )
Theorem	iotasbc2 36408*	Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( ( E! x ph /\ E! x ps ) -> ( [. ( iota x ph ) / y ]. [. ( iota x ps ) / z ]. ch <-> E. y E. z ( A. x ( ph <-> x = y ) /\ A. x ( ps <-> x = z ) /\ ch ) ) )
Theorem	pm14.12 36409*	Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( E! x ph -> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
Theorem	pm14.122a 36410*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
|- ( A e. V -> ( A. x ( ph <-> x = A ) <-> ( A. x ( ph -> x = A ) /\ [. A / x ]. ph ) ) )
Theorem	pm14.122b 36411*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
|- ( A e. V -> ( ( A. x ( ph -> x = A ) /\ [. A / x ]. ph ) <-> ( A. x ( ph -> x = A ) /\ E. x ph ) ) )
Theorem	pm14.122c 36412*	Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
|- ( A e. V -> ( A. x ( ph <-> x = A ) <-> ( A. x ( ph -> x = A ) /\ E. x ph ) ) )
Theorem	pm14.123a 36413*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
|- ( ( A e. V /\ B e. W ) -> ( A. z A. w ( ph <-> ( z = A /\ w = B ) ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) ) )
Theorem	pm14.123b 36414*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
|- ( ( A e. V /\ B e. W ) -> ( ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ E. z E. w ph ) ) )
Theorem	pm14.123c 36415*	Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
|- ( ( A e. V /\ B e. W ) -> ( A. z A. w ( ph <-> ( z = A /\ w = B ) ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ E. z E. w ph ) ) )
Theorem	pm14.18 36416	Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( E! x ph -> ( A. x ps -> [. ( iota x ph ) / x ]. ps ) )
Theorem	iotaequ 36417*	Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( iota x x = y ) = y
Theorem	iotavalb 36418*	Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5576. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )
Theorem	iotasbc5 36419*	Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( E! x ph -> ( [. ( iota x ph ) / y ]. ps <-> E. y ( y = ( iota x ph ) /\ ps ) ) )
Theorem	pm14.24 36420*	Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> A. y ( [. y / x ]. ph <-> y = ( iota x ph ) ) )
Theorem	iotavalsb 36421*	Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )
Theorem	sbiota1 36422	Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> ( A. x ( ph -> ps ) <-> [. ( iota x ph ) / x ]. ps ) )
Theorem	sbaniota 36423	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> ( E. x ( ph /\ ps ) <-> [. ( iota x ph ) / x ]. ps ) )
Theorem	eubi 36424	Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( A. x ( ph <-> ps ) -> ( E! x ph <-> E! x ps ) )
Theorem	iotasbcq 36425	Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( A. x ( ph <-> ps ) -> ( [. ( iota x ph ) / y ]. ch <-> [. ( iota x ps ) / y ]. ch ) )
21.28.5  Set Theory
Theorem	elnev 36426*	Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)
|- ( A e. _V <-> { x | -. x = A } =/= _V )
Theorem	rusbcALT 36427	A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
|- { x | x e/ x } e/ _V
Theorem	compel 36428	Equivalence between two ways of saying "is a member of the complement of A." (Contributed by Andrew Salmon, 15-Jul-2011.)
|- ( x e. ( _V \ A ) <-> -. x e. A )
Theorem	compeq 36429*	Equality between two ways of saying "the complement of A." (Contributed by Andrew Salmon, 15-Jul-2011.)
|- ( _V \ A ) = { x | -. x e. A }
Theorem	compne 36430	The complement of A is not equal to A. (Contributed by Andrew Salmon, 15-Jul-2011.)
|- ( _V \ A ) =/= A
Theorem	compab 36431	Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
|- ( _V \ { z | ph } ) = { z | -. ph }
Theorem	conss34 36432	Contrpositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
|- ( A C_ B <-> ( _V \ B ) C_ ( _V \ A ) )
Theorem	conss2 36433	Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
|- ( A C_ ( _V \ B ) <-> B C_ ( _V \ A ) )
Theorem	conss1 36434	Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
|- ( ( _V \ A ) C_ B <-> ( _V \ B ) C_ A )
Theorem	ralbidar 36435	More general form of ralbida 2865. (Contributed by Andrew Salmon, 25-Jul-2011.)
|- ( ph -> A. x e. A ph )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )
Theorem	rexbidar 36436	More general form of rexbida 2941. (Contributed by Andrew Salmon, 25-Jul-2011.)
|- ( ph -> A. x e. A ph )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )
Theorem	dropab1 36437	Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
|- ( A. x x = y -> { <. x , z >. | ph } = { <. y , z >. | ph } )
Theorem	dropab2 36438	Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
|- ( A. x x = y -> { <. z , x >. | ph } = { <. z , y >. | ph } )
Theorem	ipo0 36439	If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)
|- ( _I Po A <-> A = (/) )
Theorem	ifr0 36440	A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
|- ( _I Fr A <-> A = (/) )
Theorem	ordpss 36441	ordelpss 5470 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)
|- ( Ord B -> ( A e. B -> A C. B ) )
Theorem	fvsb 36442*	Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( E! y A F y -> ( [. ( F ` A ) / x ]. ph <-> E. x ( A. y ( A F y <-> y = x ) /\ ph ) ) )
Theorem	fveqsb 36443*	Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( x = ( F ` A ) -> ( ph <-> ps ) )   &    |- F/ x ps   =>    |- ( E! y A F y -> ( ps <-> E. x ( A. y ( A F y <-> y = x ) /\ ph ) ) )
Theorem	xpexb 36444	A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
|- ( ( A X. B ) e. _V <-> ( B X. A ) e. _V )
Theorem	trelpss 36445	An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4838, ax-reg 8107 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
|- ( ( Tr A /\ B e. A ) -> B C. A )
21.28.6  Arithmetic
Theorem	addcomgi 36446	Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)
|- ( A + B ) = ( B + A )
21.28.7  Geometry
Syntax	cplusr 36447	Introduce the operation of vector addition.
class +r
Syntax	cminusr 36448	Introduce the operation of vector subtraction.
class -r
Syntax	ctimesr 36449	Introduce the operation of scalar multiplication.
class .v
Syntax	cptdfc 36450	PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.
class PtDf ( A , B )
Syntax	crr3c 36451	RR3 is a class.
class RR3
Syntax	cline3 36452	line3 is a class.
class line3
Definition	df-addr 36453*	Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- +r = ( x e. _V , y e. _V |-> ( v e. RR |-> ( ( x ` v ) + ( y ` v ) ) ) )
Definition	df-subr 36454*	Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- -r = ( x e. _V , y e. _V |-> ( v e. RR |-> ( ( x ` v ) - ( y ` v ) ) ) )
Definition	df-mulv 36455*	Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- .v = ( x e. _V , y e. _V |-> ( v e. RR |-> ( x x. ( y ` v ) ) ) )
Theorem	addrval 36456*	Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. C /\ B e. D ) -> ( A +r B ) = ( v e. RR |-> ( ( A ` v ) + ( B ` v ) ) ) )
Theorem	subrval 36457*	Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. C /\ B e. D ) -> ( A -r B ) = ( v e. RR |-> ( ( A ` v ) - ( B ` v ) ) ) )
Theorem	mulvval 36458*	Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. C /\ B e. D ) -> ( A .v B ) = ( v e. RR |-> ( A x. ( B ` v ) ) ) )
Theorem	addrfv 36459	Vector addition at a value. The operation takes each vector A and B and forms a new vector whose values are the sum of each of the values of A and B. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. E /\ B e. D /\ C e. RR ) -> ( ( A +r B ) ` C ) = ( ( A ` C ) + ( B ` C ) ) )
Theorem	subrfv 36460	Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. E /\ B e. D /\ C e. RR ) -> ( ( A -r B ) ` C ) = ( ( A ` C ) - ( B ` C ) ) )
Theorem	mulvfv 36461	Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. E /\ B e. D /\ C e. RR ) -> ( ( A .v B ) ` C ) = ( A x. ( B ` C ) ) )
Theorem	addrfn 36462	Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. C /\ B e. D ) -> ( A +r B ) Fn RR )
Theorem	subrfn 36463	Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. C /\ B e. D ) -> ( A -r B ) Fn RR )
Theorem	mulvfn 36464	Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
|- ( ( A e. C /\ B e. D ) -> ( A .v B ) Fn RR )
Theorem	addrcom 36465	Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.)
|- ( ( A e. C /\ B e. D ) -> ( A +r B ) = ( B +r A ) )
Definition	df-ptdf 36466*	Define the predicate PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
|- PtDf ( A , B ) = ( x e. RR |-> ( ( ( x .v ( B -r A ) ) +v A ) " { 1 , 2 , 3 } ) )
Definition	df-rr3 36467	Define the set of all points RR3. We define each point A as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
|- RR3 = ( RR ^m { 1 , 2 , 3 } )
Definition	df-line3 36468*	Define the set of all lines. A line is an infinite subset of RR3 that satisfies a PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.)
|- line3 = { x e. ~P RR3 | ( 2o ~<_ x /\ A. y e. x A. z e. x ( z =/= y -> ran PtDf ( y , z ) = x ) ) }
21.29  Mathbox for Alan Sare We are sad to report the passing of long-time contributor Alan Sare (Nov. 9, 1954 - Mar. 23, 2019). Alan's first contribution to Metamath was a shorter proof for tfrlem8 7110 in 2008. He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: http://us.metamath.org/other.html#completeusersproof. His virtual deduction method is explained in the comment for wvd1 36577. Below are some excerpts from his first emails to NM in 2007: ...I have been interested in proving set theory theorems for many years for mental exercise. I enjoy it. I have used a book by Martin Zuckerman. It is informal. I am interested in completely and perfectly proving theorems. Mr. Zuckerman leaves out most of the steps of a proof, of course, like most authors do, as you have noted. A complete proof for higher theorems would require a volume of writing similar to the metamath documents. So I am frustrated when I am not capable of constructing a proof and Zuckerman leaves out steps I do not understand. I could search for the steps in other texts, but I don't do that too much. Metamath may be the answer for me.... ...If we go beyond mathematics, I believe that it is possible to write down all human knowledge in a way similar to the way you have explicated large areas of mathematics. Of course, that would be a much, much more difficult job. For example, it is possible to take a hard science like physics construct axioms based on experimental results and to cast all of physics into a collection of axioms and theorems. Maybe his has already been attempted, although I am not familiar with it. When one then moves on to the soft sciences such as social science, this job gets much more difficult. The key is: All human thought consists of logical operations on abstract objects. Usually, these logical operations are done informally. There is no reason why one cannot take any subject and explicate it and take it down to the indivisible postulates in a formal rigorous way.... ...When I read a math book or an engineering book I come across something I don't understand and I am compelled to understand it. But, often it is hopeless. I don't have the time. Or, I would have to read the same thing by multiple authors in the hope that different authors would give parts of the working proof that others have omitted. It is very inefficient. Because I have always been inclined to "get to the bottom" for a 100% fully understood proof....
21.29.1  Auxiliary theorems for the Virtual Deduction tool
Theorem	idiALT 36469	Placeholder for idi 2. Though unnecessary, this theorem is sometimes used in proofs in this mathbox for pedagogical purposes. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ph   =>    |- ph
Theorem	exbir 36470	Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 36889. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )
Theorem	3impexpbicom 36471	Version of 3impexp 1227 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
Theorem	3impexpbicomi 36472	Inference associated with 3impexpbicom 36471. Derived automatically from 3impexpbicomiVD 36894. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )
Theorem	ee02 36473	Proof of e02 36714 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.)
|- ph   &    |- ( ps -> ( ch -> th ) )   &    |- ( ph -> ( th -> ta ) )   =>    |- ( ps -> ( ch -> ta ) )
21.29.2  Supplementary unification deductions
Theorem	bi1imp 36474	Importation inference similar to imp 430, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph <-> ( ps -> ch ) )   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	bi2imp 36475	Importation inference similar to imp 430, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph <-> ( ps <-> ch ) )   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	bi3impb 36476	Similar to 3impb 1201 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ( ph /\ ( ps /\ ch ) ) <-> th )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi3impa 36477	Similar to 3impa 1200 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ( ( ph /\ ps ) /\ ch ) <-> th )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi23impib 36478	3impib 1203 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph -> ( ( ps /\ ch ) <-> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi13impib 36479	3impib 1203 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph <-> ( ( ps /\ ch ) -> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi123impib 36480	3impib 1203 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph <-> ( ( ps /\ ch ) <-> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi13impia 36481	3impia 1202 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ( ph /\ ps ) <-> ( ch -> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi123impia 36482	3impia 1202 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ( ph /\ ps ) <-> ( ch <-> th ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi33imp12 36483	3imp 1199 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph -> ( ps -> ( ch <-> th ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi23imp13 36484	3imp 1199 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph -> ( ps <-> ( ch -> th ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi13imp23 36485	3imp 1199 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph <-> ( ps -> ( ch -> th ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi13imp2 36486	Similar to 3imp 1199 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph <-> ( ps -> ( ch <-> th ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi12imp3 36487	Similar to 3imp 1199 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph <-> ( ps <-> ( ch -> th ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi23imp1 36488	Similar to 3imp 1199 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph -> ( ps <-> ( ch <-> th ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	bi123imp0 36489	Similar to 3imp 1199 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
|- ( ph <-> ( ps <-> ( ch <-> th ) ) )   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	4animp1 36490	A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.)
|- ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) )   =>    |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )
Theorem	4an31 36491	A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
|- ( ( ( ( ch /\ ps ) /\ ph ) /\ th ) -> ta )   =>    |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )
Theorem	4an4132 36492	A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
|- ( ( ( ( th /\ ch ) /\ ps ) /\ ph ) -> ta )   =>    |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )
Theorem	expcomdg 36493	Biconditional form of expcomd 439. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)
|- ( ( ph -> ( ( ps /\ ch ) -> th ) ) <-> ( ph -> ( ch -> ( ps -> th ) ) ) )
21.29.3  Conventional Metamath proofs, some derived from VD proofs
Theorem	iidn3 36494	idn3 36632 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> ( ps -> ( ch -> ch ) ) )
Theorem	ee222 36495	e222 36653 without virtual deduction connectives. Special theorem needed for the 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 ) )
Theorem	ee3bir 36496	Right-biconditional form of e3 36764 without virtual deduction connectives. Special theorem needed for the 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 ) ) )
Theorem	ee13 36497	e13 36775 without virtual deduction connectives. Special theorem needed for the 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 ) ) )
Theorem	ee121 36498	e121 36673 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 ) )
Theorem	ee122 36499	e122 36670 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 ) )
Theorem	ee333 36500	e333 36760 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 ) ) )