Theorem rspesbca 3083

Description: Existence form of rspsbca 3082. (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 3001	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
2	1	rspcev 2877	. 2 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. y e. B [ y / x ] ph )
3		cbvrexsv 2755	. 2 |- ( E. x e. B ph <-> E. y e. B [ y / x ] ph )
4	2, 3	sylibr 134	1 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph )

Syntax hints:   -> wi 4   /\ wa 104   [wsb 1785   e. wcel 2176   E.wrex 2485   [.wsbc 2998

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sbc 2999

This theorem is referenced by:  spesbc  3084