Theorem r19.37 3006

Description: Restricted quantifier version of one direction of 19.37 1966. (The other direction does not hold when is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
r19.37.1

Assertion

Ref	Expression
r19.37

Proof of Theorem r19.37

Step	Hyp	Ref	Expression
1		r19.35 3004	. 2
2		r19.37.1	. . . 4
3		ax-1 6	. . . 4
4	2, 3	ralrimi 2857	. . 3
5	4	imim1i 58	. 2
6	1, 5	sylbi 195	1

Syntax hints:  ->wi 4  F/wnf 1616  e.wcel 1818  A.wral 2807  E.wrex 2808

This theorem is referenced by:  r19.37v  3007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-12 1854

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617  df-ral 2812  df-rex 2813