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Theorem sbcrex 3328
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 2544 . 2  |-  F/_ y A
2 sbcrext 3326 . 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 setvar class
Syntax hints:    <-> wb 184   F/_wnfc 2530   E.wrex 2733   [.wsbc 3252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-v 3036  df-sbc 3253
This theorem is referenced by:  ac6sfi  7679  csbwrdg  12478  rexfiuz  13182  2sbcrex  30883  sbc2rex  30886
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