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Theorem sbcrex 3057
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcrex  |-  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem sbcrex
StepHypRef Expression
1 nfcv 2332 . 2  |-  F/_ y A
2 sbcrext 3055 . 2  |-  ( F/_ y A  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
31, 2ax-mp 5 1  |-  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   F/_wnfc 2319   E.wrex 2469   [.wsbc 2977
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978
This theorem is referenced by:  ac6sfi  6927  rexfiuz  11033
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