Theorem rspesbca 2842

Description: Existence form of rspsbca 2841. (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 2767	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
2	1	rspcev 2656	. 2 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. y e. B [ y / x ] ph )
3		cbvrexsv 2545	. 2 |- ( E. x e. B ph <-> E. y e. B [ y / x ] ph )
4	2, 3	sylibr 137	1 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   [wsb 1645   E.wrex 2307   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765

This theorem is referenced by:  spesbc  2843