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Theorem wl-cbvalnaed 31909
Description: wl-cbvalnae 31910 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)
Hypotheses
Ref Expression
wl-cbvalnaed.1  |-  F/ x ph
wl-cbvalnaed.2  |-  F/ y
ph
wl-cbvalnaed.3  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ y ps ) )
wl-cbvalnaed.4  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )
wl-cbvalnaed.5  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
wl-cbvalnaed  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )

Proof of Theorem wl-cbvalnaed
StepHypRef Expression
1 wl-cbvalnaed.1 . . . 4  |-  F/ x ph
2 wl-cbvalnaed.2 . . . 4  |-  F/ y
ph
3 wl-cbvalnaed.5 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
41, 2, 3wl-dral1d 31908 . . 3  |-  ( ph  ->  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ch ) ) )
54imp 435 . 2  |-  ( (
ph  /\  A. x  x  =  y )  ->  ( A. x ps  <->  A. y ch ) )
6 nfnae 2162 . . . 4  |-  F/ x  -.  A. x  x  =  y
71, 6nfan 2021 . . 3  |-  F/ x
( ph  /\  -.  A. x  x  =  y
)
8 wl-nfnae1 31905 . . . 4  |-  F/ y  -.  A. x  x  =  y
92, 8nfan 2021 . . 3  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
10 wl-cbvalnaed.3 . . . 4  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ y ps ) )
1110imp 435 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ y ps )
12 wl-cbvalnaed.4 . . . 4  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/ x ch ) )
1312imp 435 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ch )
143adantr 471 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
157, 9, 11, 13, 14cbv2 2123 . 2  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  ( A. x ps  <->  A. y ch ) )
165, 15pm2.61dan 805 1  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452   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-11 1930  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-cbvalnae  31910
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