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Theorem cbvex2 1988
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.)
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 . . . . 5  |-  F/ z
ph
21nfn 1840 . . . 4  |-  F/ z  -.  ph
3 cbval2.2 . . . . 5  |-  F/ w ph
43nfn 1840 . . . 4  |-  F/ w  -.  ph
5 cbval2.3 . . . . 5  |-  F/ x ps
65nfn 1840 . . . 4  |-  F/ x  -.  ps
7 cbval2.4 . . . . 5  |-  F/ y ps
87nfn 1840 . . . 4  |-  F/ y  -.  ps
9 cbval2.5 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
109notbid 294 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( -.  ph  <->  -.  ps )
)
112, 4, 6, 8, 10cbval2 1987 . . 3  |-  ( A. x A. y  -.  ph  <->  A. z A. w  -.  ps )
1211notbii 296 . 2  |-  ( -. 
A. x A. y  -.  ph  <->  -.  A. z A. w  -.  ps )
13 2exnaln 1621 . 2  |-  ( E. x E. y ph  <->  -. 
A. x A. y  -.  ph )
14 2exnaln 1621 . 2  |-  ( E. z E. w ps  <->  -. 
A. z A. w  -.  ps )
1512, 13, 143bitr4i 277 1  |-  ( E. x E. y ph  <->  E. z E. w ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368   E.wex 1587   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  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  cbvex2v  1991  2eu6OLD  2381  cbvopab  4471  cbvoprab12  6272
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