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Theorem rspesbca 3062
Description: Existence form of rspsbca 3061. (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 2980 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
21rspcev 2856 . 2  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. y  e.  B  [ y  /  x ] ph )
3 cbvrexsv 2735 . 2  |-  ( E. x  e.  B  ph  <->  E. y  e.  B  [
y  /  x ] ph )
42, 3sylibr 134 1  |-  ( ( A  e.  B  /\  [. A  /  x ]. ph )  ->  E. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   [wsb 1773    e. wcel 2160   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-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978
This theorem is referenced by:  spesbc  3063
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