Theorem bnj1294

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

	Ref	Expression
Hypotheses	bnj1294.1	|- ( ph -> A. x e. A ps )
	bnj1294.2	|- ( ph -> x e. A )
Assertion	bnj1294	|- ( ph -> ps )

Proof

Step	Hyp	Ref	Expression
1		bnj1294.1	|- ( ph -> A. x e. A ps )
2		bnj1294.2	|- ( ph -> x e. A )
3		df-ral	|- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
4		sp	|- ( A. x ( x e. A -> ps ) -> ( x e. A -> ps ) )
5	4	impcom	|- ( ( x e. A /\ A. x ( x e. A -> ps ) ) -> ps )
6	3 5	sylan2b	|- ( ( x e. A /\ A. x e. A ps ) -> ps )
7	2 1 6	syl2anc	|- ( ph -> ps )