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Theorem sb5ALTVD 34133
Description: The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Unit 20 Excercise 3.a., which is sb5 2176, found in the "Answers to Starred Exercises" on page 457 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. The same proof may also be interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sb5ALT 33701 is sb5ALTVD 34133 without virtual deductions and was automatically derived from sb5ALTVD 34133.
1::  |-  (. [ y  /  x ] ph  ->.  [ y  /  x ] ph ).
2::  |-  [ y  /  x ] x  =  y
3:1,2:  |-  (. [ y  /  x ] ph  ->.  [ y  /  x ] ( x  =  y  /\  ph ) ).
4:3:  |-  (. [ y  /  x ] ph  ->.  E. x ( x  =  y  /\  ph  ) ).
5:4:  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph )  )
6::  |-  (. E. x ( x  =  y  /\  ph )  ->.  E. x ( x  =  y  /\  ph ) ).
7::  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  ( x  =  y  /\  ph ) ).
8:7:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  ph ).
9:7:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  x  =  y ).
10:8,9:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  [ y  /  x ] ph ).
101::  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
11:101,10:  |-  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph  )
12:5,11:  |-  ( ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph  ) )  /\  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
qed:12:  |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )  )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb5ALTVD  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb5ALTVD
StepHypRef Expression
1 idn1 33764 . . . . . 6  |-  (. [
y  /  x ] ph  ->.  [ y  /  x ] ph ).
2 equsb1 2109 . . . . . 6  |-  [ y  /  x ] x  =  y
3 sban 2142 . . . . . . 7  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  <->  ( [
y  /  x ]
x  =  y  /\  [ y  /  x ] ph ) )
43simplbi2com 625 . . . . . 6  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] x  =  y  ->  [ y  /  x ] ( x  =  y  /\  ph ) ) )
51, 2, 4e10 33893 . . . . 5  |-  (. [
y  /  x ] ph  ->.  [ y  /  x ] ( x  =  y  /\  ph ) ).
6 spsbe 1748 . . . . 5  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  ph )
)
75, 6e1a 33826 . . . 4  |-  (. [
y  /  x ] ph  ->.  E. x ( x  =  y  /\  ph ) ).
87in1 33761 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
9 hbs1 2182 . . . 4  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
10 idn2 33812 . . . . . 6  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph )  ->.  ( x  =  y  /\  ph ) ).
11 simpr 459 . . . . . 6  |-  ( ( x  =  y  /\  ph )  ->  ph )
1210, 11e2 33830 . . . . 5  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph )  ->.  ph ).
13 simpl 455 . . . . . 6  |-  ( ( x  =  y  /\  ph )  ->  x  =  y )
1410, 13e2 33830 . . . . 5  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph )  ->.  x  =  y ).
15 sbequ1 1996 . . . . . 6  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
1615com12 31 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  [ y  /  x ] ph ) )
1712, 14, 16e22 33870 . . . 4  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph )  ->.  [ y  /  x ] ph ).
189, 17exinst 33823 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
198, 18pm3.2i 453 . 2  |-  ( ( [ y  /  x ] ph  ->  E. x
( x  =  y  /\  ph ) )  /\  ( E. x
( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
20 bi3 187 . . 3  |-  ( ( [ y  /  x ] ph  ->  E. x
( x  =  y  /\  ph ) )  ->  ( ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )  ->  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph ) ) ) )
2120imp 427 . 2  |-  ( ( ( [ y  /  x ] ph  ->  E. x
( x  =  y  /\  ph ) )  /\  ( E. x
( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )  -> 
( [ y  /  x ] ph  <->  E. x
( x  =  y  /\  ph ) ) )
2219, 21e0a 33982 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617   [wsb 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745  df-vd1 33760  df-vd2 33768
This theorem is referenced by: (None)
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