Theorem 19.38 1566

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 1384	. . 3 |- ( E. x ph -> A. x E. x ph )
2		hba1 1433	. . 3 |- ( A. x ps -> A. x A. x ps )
3	1, 2	hbim 1437	. 2 |- ( ( E. x ph -> A. x ps ) -> A. x ( E. x ph -> A. x ps ) )
4		19.8a 1482	. . 3 |- ( ph -> E. x ph )
5		ax-4 1400	. . 3 |- ( A. x ps -> ps )
6	4, 5	imim12i 53	. 2 |- ( ( E. x ph -> A. x ps ) -> ( ph -> ps ) )
7	3, 6	alrimih 1358	1 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )

Syntax hints:   -> wi 4   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.23t  1567  sbi2v  1772  mo3h  1953  rgenm  3323  ralm  3325