Theorem bnj1465 30915

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1465.1	|- ( x = A -> ( ph <-> ps ) )
bnj1465.2	|- ( ps -> A. x ps )
bnj1465.3	|- ( ch -> ps )

Assertion

Ref	Expression
bnj1465	|- ( ( ch /\ A e. V ) -> E. x ph )

Distinct variable groups:   x, A   x, V

Allowed substitution hints:   ph( x)   ps( x)   ch( x)

Proof of Theorem bnj1465

Step	Hyp	Ref	Expression
1		bnj1465.3	. . . 4 |- ( ch -> ps )
2	1	adantr 481	. . 3 |- ( ( ch /\ A e. V ) -> ps )
3		bnj1465.2	. . . . 5 |- ( ps -> A. x ps )
4		bnj1465.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
5	3, 4	bnj1464 30914	. . . 4 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
6	5	adantl 482	. . 3 |- ( ( ch /\ A e. V ) -> ( [. A / x ]. ph <-> ps ) )
7	2, 6	mpbird 247	. 2 |- ( ( ch /\ A e. V ) -> [. A / x ]. ph )
8	7	spesbcd 3522	1 |- ( ( ch /\ A e. V ) -> E. x ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436

This theorem is referenced by:  bnj1463  31123