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Theorem alcomw9bAUX7 29835
Description: Special case of alcom 1755 proved from ax-7v 29616. (Contributed by NM, 28-Nov-2017.)
Assertion
Ref Expression
alcomw9bAUX7  |-  ( A. x A. y ( x  =  y  /\  x  =  z )  <->  A. y A. x ( x  =  y  /\  x  =  z ) )

Proof of Theorem alcomw9bAUX7
StepHypRef Expression
1 ax7w9AUX7 29834 . 2  |-  ( A. x A. y ( x  =  y  /\  x  =  z )  ->  A. y A. x ( x  =  y  /\  x  =  z )
)
2 equcomi 1694 . . . . . 6  |-  ( x  =  y  ->  y  =  x )
32adantr 453 . . . . 5  |-  ( ( x  =  y  /\  x  =  z )  ->  y  =  x )
4 ax-8 1690 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
54imp 420 . . . . 5  |-  ( ( x  =  y  /\  x  =  z )  ->  y  =  z )
63, 5jca 520 . . . 4  |-  ( ( x  =  y  /\  x  =  z )  ->  ( y  =  x  /\  y  =  z ) )
762alimi 1570 . . 3  |-  ( A. y A. x ( x  =  y  /\  x  =  z )  ->  A. y A. x ( y  =  x  /\  y  =  z )
)
8 ax7w9AUX7 29834 . . 3  |-  ( A. y A. x ( y  =  x  /\  y  =  z )  ->  A. x A. y ( y  =  x  /\  y  =  z )
)
9 equcomi 1694 . . . . . 6  |-  ( y  =  x  ->  x  =  y )
109adantr 453 . . . . 5  |-  ( ( y  =  x  /\  y  =  z )  ->  x  =  y )
11 ax-8 1690 . . . . . 6  |-  ( y  =  x  ->  (
y  =  z  ->  x  =  z )
)
1211imp 420 . . . . 5  |-  ( ( y  =  x  /\  y  =  z )  ->  x  =  z )
1310, 12jca 520 . . . 4  |-  ( ( y  =  x  /\  y  =  z )  ->  ( x  =  y  /\  x  =  z ) )
14132alimi 1570 . . 3  |-  ( A. x A. y ( y  =  x  /\  y  =  z )  ->  A. x A. y ( x  =  y  /\  x  =  z )
)
157, 8, 143syl 19 . 2  |-  ( A. y A. x ( x  =  y  /\  x  =  z )  ->  A. x A. y ( x  =  y  /\  x  =  z )
)
161, 15impbii 182 1  |-  ( A. x A. y ( x  =  y  /\  x  =  z )  <->  A. y A. x ( x  =  y  /\  x  =  z ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   A.wal 1550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-11 1764  ax-12 1954  ax-7v 29616
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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