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Theorem cbvex2 1842
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 1571 . 2  |-  F/ z E. y ph
3 cbval2.3 . . 3  |-  F/ x ps
43nfex 1571 . 2  |-  F/ x E. w ps
5 nfv 1464 . . . . . 6  |-  F/ w  x  =  z
6 cbval2.2 . . . . . 6  |-  F/ w ph
75, 6nfan 1500 . . . . 5  |-  F/ w
( x  =  z  /\  ph )
8 nfv 1464 . . . . . 6  |-  F/ y  x  =  z
9 cbval2.4 . . . . . 6  |-  F/ y ps
108, 9nfan 1500 . . . . 5  |-  F/ y ( x  =  z  /\  ps )
11 cbval2.5 . . . . . . 7  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
1211expcom 114 . . . . . 6  |-  ( y  =  w  ->  (
x  =  z  -> 
( ph  <->  ps ) ) )
1312pm5.32d 438 . . . . 5  |-  ( y  =  w  ->  (
( x  =  z  /\  ph )  <->  ( x  =  z  /\  ps )
) )
147, 10, 13cbvex 1683 . . . 4  |-  ( E. y ( x  =  z  /\  ph )  <->  E. w ( x  =  z  /\  ps )
)
15 19.42v 1831 . . . 4  |-  ( E. y ( x  =  z  /\  ph )  <->  ( x  =  z  /\  E. y ph ) )
16 19.42v 1831 . . . 4  |-  ( E. w ( x  =  z  /\  ps )  <->  ( x  =  z  /\  E. w ps ) )
1714, 15, 163bitr3i 208 . . 3  |-  ( ( x  =  z  /\  E. y ph )  <->  ( x  =  z  /\  E. w ps ) )
18 pm5.32 441 . . 3  |-  ( ( x  =  z  -> 
( E. y ph  <->  E. w ps ) )  <-> 
( ( x  =  z  /\  E. y ph )  <->  ( x  =  z  /\  E. w ps ) ) )
1917, 18mpbir 144 . 2  |-  ( x  =  z  ->  ( E. y ph  <->  E. w ps ) )
202, 4, 19cbvex 1683 1  |-  ( E. x E. y ph  <->  E. z E. w ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   F/wnf 1392   E.wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  cbvex2v  1844  cbvopab  3884  cbvoprab12  5679
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