Theorem rspesbca 2898

Description: Existence form of rspsbca 2897. (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.

Step	Hyp	Ref	Expression
1		dfsbcq2 2818	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
2	1	rspcev 2701	. 2 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. y e. B [ y / x ] ph )
3		cbvrexsv 2589	. 2 |- ( E. x e. B ph <-> E. y e. B [ y / x ] ph )
4	2, 3	sylibr 132	1 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   [wsb 1685   E.wrex 2349   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816

This theorem is referenced by:  spesbc  2899