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Theorem bj-cbval2v 32595
Description: Version of cbval2 1987 with a dv condition, which does not require ax-13 1955. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbval2v.1  |-  F/ z
ph
bj-cbval2v.2  |-  F/ w ph
bj-cbval2v.3  |-  F/ x ps
bj-cbval2v.4  |-  F/ y ps
bj-cbval2v.5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
bj-cbval2v  |-  ( A. x A. y ph  <->  A. z A. w ps )
Distinct variable group:    x, y, z, w
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem bj-cbval2v
StepHypRef Expression
1 bj-cbval2v.1 . . 3  |-  F/ z
ph
21nfal 1885 . 2  |-  F/ z A. y ph
3 bj-cbval2v.3 . . 3  |-  F/ x ps
43nfal 1885 . 2  |-  F/ x A. w ps
5 nfv 1674 . . . . . 6  |-  F/ w  x  =  z
6 bj-cbval2v.2 . . . . . 6  |-  F/ w ph
75, 6nfim 1858 . . . . 5  |-  F/ w
( x  =  z  ->  ph )
8 nfv 1674 . . . . . 6  |-  F/ y  x  =  z
9 bj-cbval2v.4 . . . . . 6  |-  F/ y ps
108, 9nfim 1858 . . . . 5  |-  F/ y ( x  =  z  ->  ps )
11 bj-cbval2v.5 . . . . . . 7  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
1211expcom 435 . . . . . 6  |-  ( y  =  w  ->  (
x  =  z  -> 
( ph  <->  ps ) ) )
1312pm5.74d 247 . . . . 5  |-  ( y  =  w  ->  (
( x  =  z  ->  ph )  <->  ( x  =  z  ->  ps )
) )
147, 10, 13bj-cbvalv 32589 . . . 4  |-  ( A. y ( x  =  z  ->  ph )  <->  A. w
( x  =  z  ->  ps ) )
15 19.21v 1921 . . . 4  |-  ( A. y ( x  =  z  ->  ph )  <->  ( x  =  z  ->  A. y ph ) )
16 19.21v 1921 . . . 4  |-  ( A. w ( x  =  z  ->  ps )  <->  ( x  =  z  ->  A. w ps ) )
1714, 15, 163bitr3i 275 . . 3  |-  ( ( x  =  z  ->  A. y ph )  <->  ( x  =  z  ->  A. w ps ) )
1817pm5.74ri 246 . 2  |-  ( x  =  z  ->  ( A. y ph  <->  A. w ps ) )
192, 4, 18bj-cbvalv 32589 1  |-  ( A. x A. y ph  <->  A. z A. w ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368   F/wnf 1590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  bj-cbvex2v  32596  bj-cbval2vv  32597
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