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Theorem wl-dral1d 31934
Description: A version of dral1 2174 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 
x  =  y,  A. x x  =  y or  ph antecedent. wl-equsal1i 31946 and nfdi 2019 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.)
Hypotheses
Ref Expression
wl-dral1d.1  |-  F/ x ph
wl-dral1d.2  |-  F/ y
ph
wl-dral1d.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
wl-dral1d  |-  ( ph  ->  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ch ) ) )

Proof of Theorem wl-dral1d
StepHypRef Expression
1 wl-dral1d.3 . . . . . . . 8  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
21com12 31 . . . . . . 7  |-  ( x  =  y  ->  ( ph  ->  ( ps  <->  ch )
) )
32pm5.74d 255 . . . . . 6  |-  ( x  =  y  ->  (
( ph  ->  ps )  <->  (
ph  ->  ch ) ) )
43sps 1963 . . . . 5  |-  ( A. x  x  =  y  ->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
54dral1 2174 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x (
ph  ->  ps )  <->  A. y
( ph  ->  ch )
) )
6 wl-dral1d.1 . . . . 5  |-  F/ x ph
7619.21 2007 . . . 4  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
8 wl-dral1d.2 . . . . 5  |-  F/ y
ph
9819.21 2007 . . . 4  |-  ( A. y ( ph  ->  ch )  <->  ( ph  ->  A. y ch ) )
105, 7, 93bitr3g 295 . . 3  |-  ( A. x  x  =  y  ->  ( ( ph  ->  A. x ps )  <->  ( ph  ->  A. y ch )
) )
1110pm5.74rd 256 . 2  |-  ( A. x  x  =  y  ->  ( ph  ->  ( A. x ps  <->  A. y ch ) ) )
1211com12 31 1  |-  ( ph  ->  ( A. x  x  =  y  ->  ( A. x ps  <->  A. y ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450   F/wnf 1675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by:  wl-cbvalnaed  31935
  Copyright terms: Public domain W3C validator