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Theorem rspesbca 3241
Description: Existence form of rspsbca 3240. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
rspesbca  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. x  e.  B  ph )
Distinct variable group:    x, B
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem rspesbca
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3164 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
21rspcev 3052 . 2  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. y  e.  B  [ y  /  x ] ph )
3 cbvrexsv 2944 . 2  |-  ( E. x  e.  B  ph  <->  E. y  e.  B  [
y  /  x ] ph )
42, 3sylibr 204 1  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   [wsb 1658    e. wcel 1725   E.wrex 2706   [.wsbc 3161
This theorem is referenced by:  spesbc  3242  indexfi  7414  indexdom  26436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-sbc 3162
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