Theorem r19.36v 2949

Description: Restricted quantifier version of one direction of 19.36 2054. (The other direction holds iff A is nonempty, see r19.36zv 3881.) (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
r19.36v	|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem r19.36v

Step	Hyp	Ref	Expression
1		r19.35 2948	. 2 |- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
2		idd 25	. . . 4 |- ( x e. A -> ( ps -> ps ) )
3	2	rexlimiv 2884	. . 3 |- ( E. x e. A ps -> ps )
4	3	imim2i 16	. 2 |- ( ( A. x e. A ph -> E. x e. A ps ) -> ( A. x e. A ph -> ps ) )
5	1, 4	sylbi 200	1 |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   e. wcel 1897   A.wral 2748   E.wrex 2749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-ral 2753  df-rex 2754

This theorem is referenced by:  iinss  4342  uniimadom  8994  hashgt12el  12627