Theorem 19.41h 1707

Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1708 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 1653	. . 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 1615	. . . 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 1633	. . 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 1370   E.wex 1514

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.42h  1709  sbh  1798  sbidm  1873  19.41v  1925  2exeu  2145