Theorem r19.29imd 3074

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.)

Hypotheses

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

Assertion

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

Proof of Theorem r19.29imd

Step	Hyp	Ref	Expression
1		r19.29imd.1	. . 3 |- ( ph -> E. x e. A ps )
2		r19.29imd.2	. . 3 |- ( ph -> A. x e. A ( ps -> ch ) )
3		r19.29r 3073	. . 3 |- ( ( E. x e. A ps /\ A. x e. A ( ps -> ch ) ) -> E. x e. A ( ps /\ ( ps -> ch ) ) )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> E. x e. A ( ps /\ ( ps -> ch ) ) )
5		abai 836	. . 3 |- ( ( ps /\ ch ) <-> ( ps /\ ( ps -> ch ) ) )
6	5	rexbii 3041	. 2 |- ( E. x e. A ( ps /\ ch ) <-> E. x e. A ( ps /\ ( ps -> ch ) ) )
7	4, 6	sylibr 224	1 |- ( ph -> E. x e. A ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wral 2912   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  psgndif  19948  neik0pk1imk0  38345