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Theorem ssoprab2 6122
 Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4472. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
ssoprab2

Proof of Theorem ssoprab2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 id 20 . . . . . . . . . 10
21anim2d 549 . . . . . . . . 9
32alimi 1568 . . . . . . . 8
4 exim 1584 . . . . . . . 8
53, 4syl 16 . . . . . . 7
65alimi 1568 . . . . . 6
7 exim 1584 . . . . . 6
86, 7syl 16 . . . . 5
98alimi 1568 . . . 4
10 exim 1584 . . . 4
119, 10syl 16 . . 3
1211ss2abdv 3408 . 2
13 df-oprab 6077 . 2
14 df-oprab 6077 . 2
1512, 13, 143sstr4g 3381 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549  wex 1550   wceq 1652  cab 2421   wss 3312  cop 3809  coprab 6074 This theorem is referenced by:  ssoprab2b  6123 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-in 3319  df-ss 3326  df-oprab 6077
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