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Theorem cbvrexfw 2696
Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2698 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvrexfw.1  |-  F/_ x A
cbvrexfw.2  |-  F/_ y A
cbvrexfw.3  |-  F/ y
ph
cbvrexfw.4  |-  F/ x ps
cbvrexfw.5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrexfw  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x, y)

Proof of Theorem cbvrexfw
StepHypRef Expression
1 cbvrexfw.2 . . . . 5  |-  F/_ y A
21nfcri 2313 . . . 4  |-  F/ y  x  e.  A
3 cbvrexfw.3 . . . 4  |-  F/ y
ph
42, 3nfan 1565 . . 3  |-  F/ y ( x  e.  A  /\  ph )
5 cbvrexfw.1 . . . . 5  |-  F/_ x A
65nfcri 2313 . . . 4  |-  F/ x  y  e.  A
7 cbvrexfw.4 . . . 4  |-  F/ x ps
86, 7nfan 1565 . . 3  |-  F/ x
( y  e.  A  /\  ps )
9 eleq1w 2238 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
10 cbvrexfw.5 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
119, 10anbi12d 473 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  /\  ph )  <->  ( y  e.  A  /\  ps )
) )
124, 8, 11cbvexv1 1752 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. y ( y  e.  A  /\  ps )
)
13 df-rex 2461 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
14 df-rex 2461 . 2  |-  ( E. y  e.  A  ps  <->  E. y ( y  e.  A  /\  ps )
)
1512, 13, 143bitr4i 212 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   F/wnf 1460   E.wex 1492    e. wcel 2148   F/_wnfc 2306   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461
This theorem is referenced by:  nnwofdc  12039
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