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Theorem r19.12sn 3874
 Description: Special case of r19.12 2821 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
r19.12sn.1
Assertion
Ref Expression
r19.12sn
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem r19.12sn
StepHypRef Expression
1 r19.12sn.1 . 2
2 sbcralg 3237 . . 3
3 rexsns 3847 . . 3
4 rexsns 3847 . . . 4
54ralbidv 2727 . . 3
62, 3, 53bitr4d 278 . 2
71, 6ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wb 178   wcel 1726  wral 2707  wrex 2708  cvv 2958  wsbc 3163  csn 3816 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-sn 3822
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