Theorem r19.29imd

Description: Theorem 19.29 of Margaris p. 90 with an implication in the hypothesis containing the generalization, deduction version. (Contributed by AV, 19-Jan-2019)

	Ref	Expression
Hypotheses	r19.29imd.1	|- ( ph -> E. x e. A ps )
	r19.29imd.2	|- ( ph -> A. x e. A ( ps -> ch ) )
Assertion	r19.29imd	|- ( ph -> E. x e. A ( ps /\ ch ) )

Proof

Step	Hyp	Ref	Expression
1		r19.29imd.1	|- ( ph -> E. x e. A ps )
2		r19.29imd.2	|- ( ph -> A. x e. A ( ps -> ch ) )
3		r19.29r	|- ( ( E. x e. A ps /\ A. x e. A ( ps -> ch ) ) -> E. x e. A ( ps /\ ( ps -> ch ) ) )
4	1 2 3	syl2anc	|- ( ph -> E. x e. A ( ps /\ ( ps -> ch ) ) )
5		abai	|- ( ( ps /\ ch ) <-> ( ps /\ ( ps -> ch ) ) )
6	5	rexbii	|- ( E. x e. A ( ps /\ ch ) <-> E. x e. A ( ps /\ ( ps -> ch ) ) )
7	4 6	sylibr	|- ( ph -> E. x e. A ( ps /\ ch ) )