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Theorem ax11-pm2 34810
Description: Proof of ax-11 1847 from the standard axioms of predicate calculus, similar to PM's proof of alcom 1850 (PM*11.2). This proof requires that  x and  y be distinct. Axiom ax-11 1847 is used in the proof only through nfal 1952, nfsb 2186, sbal 2208, sb8 2169. See also ax11-pm 34806. (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 2097 . . . . . 6  |-  ( A. x A. y ph  ->  [ z  /  x ] [ t  /  y ] ph )
21gen2 1624 . . . . 5  |-  A. t A. z ( A. x A. y ph  ->  [ z  /  x ] [
t  /  y ]
ph )
3 nfv 1712 . . . . . . . 8  |-  F/ t
ph
43nfal 1952 . . . . . . 7  |-  F/ t A. y ph
54nfal 1952 . . . . . 6  |-  F/ t A. x A. y ph
6 nfv 1712 . . . . . . . 8  |-  F/ z
ph
76nfal 1952 . . . . . . 7  |-  F/ z A. y ph
87nfal 1952 . . . . . 6  |-  F/ z A. x A. y ph
95, 82stdpc5 34803 . . . . 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 2186 . . . . . 6  |-  F/ z [ t  /  y ] ph
1211sb8 2169 . . . . 5  |-  ( A. x [ t  /  y ] ph  <->  A. z [ z  /  x ] [
t  /  y ]
ph )
1312albii 1645 . . . 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 2208 . . . 4  |-  ( [ t  /  y ] A. x ph  <->  A. x [ t  /  y ] ph )
1615albii 1645 . . 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 1952 . . 3  |-  F/ t A. x ph
1918sb8 2169 . 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 1396   [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-11 1847  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
This theorem is referenced by: (None)
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