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Theorem ax11-pm2 32646
Description: Proof of ax-11 1782 from the standard axioms of predicate calculus, similar to PM's proof of alcom 1785 (PM*11.2). This proof requires that  x and  y be distinct. Axiom ax-11 1782 is used in the proof only through nfal 1882, nfsb 2153, sbal 2181, sb8 2131. See also ax11-pm 32642. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax11-pm2  |-  ( A. x A. y ph  ->  A. y A. x ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax11-pm2
Dummy variables  z 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2stdpc4 2052 . . . . . 6  |-  ( A. x A. y ph  ->  [ z  /  x ] [ t  /  y ] ph )
21gen2 1593 . . . . 5  |-  A. t A. z ( A. x A. y ph  ->  [ z  /  x ] [
t  /  y ]
ph )
3 nfv 1674 . . . . . . . 8  |-  F/ t
ph
43nfal 1882 . . . . . . 7  |-  F/ t A. y ph
54nfal 1882 . . . . . 6  |-  F/ t A. x A. y ph
6 nfv 1674 . . . . . . . 8  |-  F/ z
ph
76nfal 1882 . . . . . . 7  |-  F/ z A. y ph
87nfal 1882 . . . . . 6  |-  F/ z A. x A. y ph
95, 82stdpc5 32639 . . . . 5  |-  ( A. t A. z ( A. x A. y ph  ->  [ z  /  x ] [ t  /  y ] ph )  ->  ( A. x A. y ph  ->  A. t A. z [ z  /  x ] [ t  /  y ] ph ) )
102, 9ax-mp 5 . . . 4  |-  ( A. x A. y ph  ->  A. t A. z [ z  /  x ] [ t  /  y ] ph )
116nfsb 2153 . . . . . 6  |-  F/ z [ t  /  y ] ph
1211sb8 2131 . . . . 5  |-  ( A. x [ t  /  y ] ph  <->  A. z [ z  /  x ] [
t  /  y ]
ph )
1312albii 1611 . . . 4  |-  ( A. t A. x [ t  /  y ] ph  <->  A. t A. z [ z  /  x ] [ t  /  y ] ph )
1410, 13sylibr 212 . . 3  |-  ( A. x A. y ph  ->  A. t A. x [
t  /  y ]
ph )
15 sbal 2181 . . . 4  |-  ( [ t  /  y ] A. x ph  <->  A. x [ t  /  y ] ph )
1615albii 1611 . . 3  |-  ( A. t [ t  /  y ] A. x ph  <->  A. t A. x [ t  / 
y ] ph )
1714, 16sylibr 212 . 2  |-  ( A. x A. y ph  ->  A. t [ t  / 
y ] A. x ph )
183nfal 1882 . . 3  |-  F/ t A. x ph
1918sb8 2131 . 2  |-  ( A. y A. x ph  <->  A. t [ t  /  y ] A. x ph )
2017, 19sylibr 212 1  |-  ( A. x A. y ph  ->  A. y A. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368   [wsb 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by: (None)
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