Theorem 19.41h 1696

Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1697 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.41h.1	|- ( ps -> A. x ps )

Assertion

Ref	Expression
19.41h	|- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )

Proof of Theorem 19.41h

Step	Hyp	Ref	Expression
1		19.40 1642	. . 3 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
2		19.41h.1	. . . . 5 |- ( ps -> A. x ps )
3		id 19	. . . . 5 |- ( ps -> ps )
4	2, 3	exlimih 1604	. . . 4 |- ( E. x ps -> ps )
5	4	anim2i 342	. . 3 |- ( ( E. x ph /\ E. x ps ) -> ( E. x ph /\ ps ) )
6	1, 5	syl 14	. 2 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ ps ) )
7		pm3.21 264	. . . 4 |- ( ps -> ( ph -> ( ph /\ ps ) ) )
8	2, 7	eximdh 1622	. . 3 |- ( ps -> ( E. x ph -> E. x ( ph /\ ps ) ) )
9	8	impcom 125	. 2 |- ( ( E. x ph /\ ps ) -> E. x ( ph /\ ps ) )
10	6, 9	impbii 126	1 |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.42h  1698  sbh  1787  sbidm  1862  19.41v  1914  2exeu  2134