Theorem bnj1465

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

	Ref	Expression
Hypotheses	bnj1465.1	|- ( x = A -> ( ph <-> ps ) )
	bnj1465.2	|- ( ps -> A. x ps )
	bnj1465.3	|- ( ch -> ps )
Assertion	bnj1465	|- ( ( ch /\ A e. V ) -> E. x ph )

Proof

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