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Theorem cbvrexcsf 3144
Description: A more general version of cbvrexf 2759 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
cbvralcsf.1  |-  F/_ y A
cbvralcsf.2  |-  F/_ x B
cbvralcsf.3  |-  F/ y
ph
cbvralcsf.4  |-  F/ x ps
cbvralcsf.5  |-  ( x  =  y  ->  A  =  B )
cbvralcsf.6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrexcsf  |-  ( E. x  e.  A  ph  <->  E. y  e.  B  ps )

Proof of Theorem cbvrexcsf
StepHypRef Expression
1 cbvralcsf.1 . . . 4  |-  F/_ y A
2 cbvralcsf.2 . . . 4  |-  F/_ x B
3 cbvralcsf.3 . . . . 5  |-  F/ y
ph
43nfn 1765 . . . 4  |-  F/ y  -.  ph
5 cbvralcsf.4 . . . . 5  |-  F/ x ps
65nfn 1765 . . . 4  |-  F/ x  -.  ps
7 cbvralcsf.5 . . . 4  |-  ( x  =  y  ->  A  =  B )
8 cbvralcsf.6 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
98notbid 285 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
101, 2, 4, 6, 7, 9cbvralcsf 3143 . . 3  |-  ( A. x  e.  A  -.  ph  <->  A. y  e.  B  -.  ps )
1110notbii 287 . 2  |-  ( -. 
A. x  e.  A  -.  ph  <->  -.  A. y  e.  B  -.  ps )
12 dfrex2 2556 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
13 dfrex2 2556 . 2  |-  ( E. y  e.  B  ps  <->  -. 
A. y  e.  B  -.  ps )
1411, 12, 133bitr4i 268 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   F/wnf 1531    = wceq 1623   F/_wnfc 2406   A.wral 2543   E.wrex 2544
This theorem is referenced by:  cbvrexv2  3148
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  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-sbc 2992  df-csb 3082
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