Theorem r19.37 3086

Description: Restricted quantifier version of one direction of 19.37 2100. (The other direction does not hold when A is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
r19.37.1	|- F/ x ph

Assertion

Ref	Expression
r19.37	|- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) )

Proof of Theorem r19.37

Step	Hyp	Ref	Expression
1		r19.35 3084	. 2 |- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
2		r19.37.1	. . . 4 |- F/ x ph
3		ax-1 6	. . . 4 |- ( ph -> ( x e. A -> ph ) )
4	2, 3	ralrimi 2957	. . 3 |- ( ph -> A. x e. A ph )
5	4	imim1i 63	. 2 |- ( ( A. x e. A ph -> E. x e. A ps ) -> ( ph -> E. x e. A ps ) )
6	1, 5	sylbi 207	1 |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) )

Syntax hints:   -> wi 4   F/wnf 1708   e. wcel 1990   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  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  r19.37v  3087  ss2iundf  37951