Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.37 Unicode version

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
StepHypRef Expression
1 r19.35 3004 . 2
2 r19.37.1 . . . 4
3 ax-1 6 . . . 4
42, 3ralrimi 2857 . . 3
54imim1i 58 . 2
61, 5sylbi 195 1
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator