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Theorem rspesbca 3071
Description: Existence form of rspsbca 3070. (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 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
21rspcev 2884 . 2  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. y  e.  B  [ y  /  x ] ph )
3 cbvrexsv 2776 . 2  |-  ( E. x  e.  B  ph  <->  E. y  e.  B  [
y  /  x ] ph )
42, 3sylibr 203 1  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   [wsb 1629    e. wcel 1684   E.wrex 2544   [.wsbc 2991
This theorem is referenced by:  spesbc  3072  indexfi  7163  indexdom  26413
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-v 2790  df-sbc 2992
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