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Theorem r19.12sn 3466
 Description: Special case of r19.12 2467 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)
Assertion
Ref Expression
r19.12sn
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)   ()   (,)

Proof of Theorem r19.12sn
StepHypRef Expression
1 sbcralg 2893 . 2
2 rexsns 3440 . 2
3 rexsns 3440 . . 3
43ralbii 2373 . 2
51, 2, 43bitr4g 221 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 103   wcel 1434  wral 2349  wrex 2350  wsbc 2816  csn 3406 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-sn 3412 This theorem is referenced by: (None)
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