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Theorem wl-equsald 31916
Description: Deduction version of equsal 2138. (Contributed by Wolf Lammen, 27-Jul-2019.)
Hypotheses
Ref Expression
wl-equsald.1  |-  F/ x ph
wl-equsald.2  |-  ( ph  ->  F/ x ch )
wl-equsald.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
wl-equsald  |-  ( ph  ->  ( A. x ( x  =  y  ->  ps )  <->  ch ) )

Proof of Theorem wl-equsald
StepHypRef Expression
1 wl-equsald.2 . . 3  |-  ( ph  ->  F/ x ch )
2 19.23t 2001 . . 3  |-  ( F/ x ch  ->  ( A. x ( x  =  y  ->  ch )  <->  ( E. x  x  =  y  ->  ch )
) )
31, 2syl 17 . 2  |-  ( ph  ->  ( A. x ( x  =  y  ->  ch )  <->  ( E. x  x  =  y  ->  ch ) ) )
4 wl-equsald.1 . . 3  |-  F/ x ph
5 wl-equsald.3 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
65pm5.74d 255 . . 3  |-  ( ph  ->  ( ( x  =  y  ->  ps )  <->  ( x  =  y  ->  ch ) ) )
74, 6albid 1973 . 2  |-  ( ph  ->  ( A. x ( x  =  y  ->  ps )  <->  A. x ( x  =  y  ->  ch ) ) )
8 ax6e 2104 . . . 4  |-  E. x  x  =  y
98a1bi 343 . . 3  |-  ( ch  <->  ( E. x  x  =  y  ->  ch )
)
109a1i 11 . 2  |-  ( ph  ->  ( ch  <->  ( E. x  x  =  y  ->  ch ) ) )
113, 7, 103bitr4d 293 1  |-  ( ph  ->  ( A. x ( x  =  y  ->  ps )  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1452   E.wex 1673   F/wnf 1677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678
This theorem is referenced by:  wl-equsal  31917  wl-equsal1t  31918  wl-sb6rft  31921
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