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Theorem rspesbca 3360
Description: Existence form of rspsbca 3359. (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 3282 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
21rspcev 3162 . 2  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. y  e.  B  [ y  /  x ] ph )
3 cbvrexsv 3043 . 2  |-  ( E. x  e.  B  ph  <->  E. y  e.  B  [
y  /  x ] ph )
42, 3sylibr 217 1  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. x  e.  B  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375   [wsb 1808    e. wcel 1898   E.wrex 2750   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-rex 2755  df-v 3059  df-sbc 3280
This theorem is referenced by:  spesbc  3361  indexfi  7913  indexdom  32107
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