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Theorem wl-equsal1i 29849
Description: The antecedent  x  =  y is irrelevant, if one or both setvar variables are not free in  ph. (Contributed by Wolf Lammen, 1-Sep-2018.)
Hypotheses
Ref Expression
wl-equsal1i.1  |-  ( F/ x ph  \/  F/ y ph )
wl-equsal1i.2  |-  ( x  =  y  ->  ph )
Assertion
Ref Expression
wl-equsal1i  |-  ph

Proof of Theorem wl-equsal1i
StepHypRef Expression
1 wl-equsal1i.1 . 2  |-  ( F/ x ph  \/  F/ y ph )
2 wl-equsal1i.2 . . 3  |-  ( x  =  y  ->  ph )
32gen2 1602 . 2  |-  A. x A. y ( x  =  y  ->  ph )
4 sp 1808 . . . . 5  |-  ( A. y A. x ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  ph )
)
54alcoms 1792 . . . 4  |-  ( A. x A. y ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  ph )
)
6 wl-equsal1t 29847 . . . 4  |-  ( F/ x ph  ->  ( A. x ( x  =  y  ->  ph )  <->  ph ) )
75, 6syl5ib 219 . . 3  |-  ( F/ x ph  ->  ( A. x A. y ( x  =  y  ->  ph )  ->  ph )
)
8 wl-equsalcom 29848 . . . . 5  |-  ( A. y ( y  =  x  ->  ph )  <->  A. y
( x  =  y  ->  ph ) )
9 wl-equsal1t 29847 . . . . . 6  |-  ( F/ y ph  ->  ( A. y ( y  =  x  ->  ph )  <->  ph ) )
109biimpd 207 . . . . 5  |-  ( F/ y ph  ->  ( A. y ( y  =  x  ->  ph )  ->  ph ) )
118, 10syl5bir 218 . . . 4  |-  ( F/ y ph  ->  ( A. y ( x  =  y  ->  ph )  ->  ph ) )
1211spsd 1816 . . 3  |-  ( F/ y ph  ->  ( A. x A. y ( x  =  y  ->  ph )  ->  ph )
)
137, 12jaoi 379 . 2  |-  ( ( F/ x ph  \/  F/ y ph )  -> 
( A. x A. y ( x  =  y  ->  ph )  ->  ph ) )
141, 3, 13mp2 9 1  |-  ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368   A.wal 1377   F/wnf 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by: (None)
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