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Theorem cbvex2 2039
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-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
cbvex2  |-  ( E. x E. y ph  <->  E. z E. 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 cbvex2
StepHypRef Expression
1 cbval2.1 . . 3  |-  F/ z
ph
21nfex 1855 . 2  |-  F/ z E. y ph
3 cbval2.3 . . 3  |-  F/ x ps
43nfex 1855 . 2  |-  F/ x E. w ps
5 nfv 1626 . . . . . 6  |-  F/ w  x  =  z
6 cbval2.2 . . . . . 6  |-  F/ w ph
75, 6nfan 1836 . . . . 5  |-  F/ w
( x  =  z  /\  ph )
8 nfv 1626 . . . . . 6  |-  F/ y  x  =  z
9 cbval2.4 . . . . . 6  |-  F/ y ps
108, 9nfan 1836 . . . . 5  |-  F/ y ( x  =  z  /\  ps )
11 cbval2.5 . . . . . . 7  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
1211expcom 425 . . . . . 6  |-  ( y  =  w  ->  (
x  =  z  -> 
( ph  <->  ps ) ) )
1312pm5.32d 621 . . . . 5  |-  ( y  =  w  ->  (
( x  =  z  /\  ph )  <->  ( x  =  z  /\  ps )
) )
147, 10, 13cbvex 2020 . . . 4  |-  ( E. y ( x  =  z  /\  ph )  <->  E. w ( x  =  z  /\  ps )
)
15 19.42v 1917 . . . 4  |-  ( E. y ( x  =  z  /\  ph )  <->  ( x  =  z  /\  E. y ph ) )
16 19.42v 1917 . . . 4  |-  ( E. w ( x  =  z  /\  ps )  <->  ( x  =  z  /\  E. w ps ) )
1714, 15, 163bitr3i 267 . . 3  |-  ( ( x  =  z  /\  E. y ph )  <->  ( x  =  z  /\  E. w ps ) )
18 pm5.32 618 . . 3  |-  ( ( x  =  z  -> 
( E. y ph  <->  E. w ps ) )  <-> 
( ( x  =  z  /\  E. y ph )  <->  ( x  =  z  /\  E. w ps ) ) )
1917, 18mpbir 201 . 2  |-  ( x  =  z  ->  ( E. y ph  <->  E. w ps ) )
202, 4, 19cbvex 2020 1  |-  ( E. x E. y ph  <->  E. z E. w ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547   F/wnf 1550
This theorem is referenced by:  cbvex2v  2041  2eu6  2323  cbvopab  4217  cbvoprab12  6085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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