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Theorem cbval2 1944
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
cbval2.1  |-  F/ z
ph
cbval2.2  |-  F/ w ph
cbval2.3  |-  F/ x ps
cbval2.4  |-  F/ y ps
cbval2.5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbval2  |-  ( A. x A. y ph  <->  A. z A. w ps )
Distinct variable groups:    x, y    y, z    x, w    z, w
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem cbval2
StepHypRef Expression
1 cbval2.1 . . 3  |-  F/ z
ph
21nfal 1766 . 2  |-  F/ z A. y ph
3 cbval2.3 . . 3  |-  F/ x ps
43nfal 1766 . 2  |-  F/ x A. w ps
5 nfv 1605 . . . . . 6  |-  F/ w  x  =  z
6 cbval2.2 . . . . . 6  |-  F/ w ph
75, 6nfan 1771 . . . . 5  |-  F/ w
( x  =  z  /\  ph )
8 nfv 1605 . . . . . 6  |-  F/ y  x  =  z
9 cbval2.4 . . . . . 6  |-  F/ y ps
108, 9nfan 1771 . . . . 5  |-  F/ y ( x  =  z  /\  ps )
11 cbval2.5 . . . . . . 7  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
1211expcom 424 . . . . . 6  |-  ( y  =  w  ->  (
x  =  z  -> 
( ph  <->  ps ) ) )
1312pm5.32d 620 . . . . 5  |-  ( y  =  w  ->  (
( x  =  z  /\  ph )  <->  ( x  =  z  /\  ps )
) )
147, 10, 13cbval 1924 . . . 4  |-  ( A. y ( x  =  z  /\  ph )  <->  A. w ( x  =  z  /\  ps )
)
15 19.28v 1836 . . . 4  |-  ( A. y ( x  =  z  /\  ph )  <->  ( x  =  z  /\  A. y ph ) )
16 19.28v 1836 . . . 4  |-  ( A. w ( x  =  z  /\  ps )  <->  ( x  =  z  /\  A. w ps ) )
1714, 15, 163bitr3i 266 . . 3  |-  ( ( x  =  z  /\  A. y ph )  <->  ( x  =  z  /\  A. w ps ) )
18 pm5.32 617 . . 3  |-  ( ( x  =  z  -> 
( A. y ph  <->  A. w ps ) )  <-> 
( ( x  =  z  /\  A. y ph )  <->  ( x  =  z  /\  A. w ps ) ) )
1917, 18mpbir 200 . 2  |-  ( x  =  z  ->  ( A. y ph  <->  A. w ps ) )
202, 4, 19cbval 1924 1  |-  ( A. x A. y ph  <->  A. z A. w ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   F/wnf 1531
This theorem is referenced by:  cbval2v  1946  2mo  2221  2eu6  2228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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