Theorem 19.38 1607

Description: Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.38	|- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )

Proof of Theorem 19.38

Step	Hyp	Ref	Expression
1		hbe1 1425	. . 3 |- ( E. x ph -> A. x E. x ph )
2		hba1 1474	. . 3 |- ( A. x ps -> A. x A. x ps )
3	1, 2	hbim 1478	. 2 |- ( ( E. x ph -> A. x ps ) -> A. x ( E. x ph -> A. x ps ) )
4		19.8a 1523	. . 3 |- ( ph -> E. x ph )
5		ax-4 1441	. . 3 |- ( A. x ps -> ps )
6	4, 5	imim12i 58	. 2 |- ( ( E. x ph -> A. x ps ) -> ( ph -> ps ) )
7	3, 6	alrimih 1399	1 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )

Syntax hints:   -> wi 4   A.wal 1283   E.wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  19.23t  1608  sbi2v  1815  mo3h  1996  rgenm  3360  ralm  3362