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Theorem r19.12sn 4095
Description: Special case of r19.12 2983 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)
Assertion
Ref Expression
r19.12sn
Distinct variable groups:   , ,   ,

Proof of Theorem r19.12sn
StepHypRef Expression
1 sbcralg 3411 . 2
2 rexsns 4062 . 2
3 rexsns 4062 . . 3
43ralbii 2888 . 2
51, 2, 43bitr4g 288 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  e.wcel 1818  A.wral 2807  E.wrex 2808  [.wsbc 3327  {csn 4029
This theorem is referenced by:  intimasn  37771
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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-sn 4030
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