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Theorem ssoprab2b 6354
 Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4779. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
ssoprab2b

Proof of Theorem ssoprab2b
StepHypRef Expression
1 nfoprab1 6346 . . . 4
2 nfoprab1 6346 . . . 4
31, 2nfss 3496 . . 3
4 nfoprab2 6347 . . . . 5
5 nfoprab2 6347 . . . . 5
64, 5nfss 3496 . . . 4
7 nfoprab3 6348 . . . . . 6
8 nfoprab3 6348 . . . . . 6
97, 8nfss 3496 . . . . 5
10 ssel 3497 . . . . . 6
11 oprabid 6323 . . . . . 6
12 oprabid 6323 . . . . . 6
1310, 11, 123imtr3g 269 . . . . 5
149, 13alrimi 1877 . . . 4
156, 14alrimi 1877 . . 3
163, 15alrimi 1877 . 2
17 ssoprab2 6353 . 2
1816, 17impbii 188 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  e.wcel 1818  C_wss 3475  <.cop 4035  {coprab 6297 This theorem is referenced by:  eqoprab2b  6355 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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-oprab 6300
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