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Theorem opeqpr 4445
 Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
opeqpr.1
opeqpr.2
opeqpr.3
opeqpr.4
Assertion
Ref Expression
opeqpr

Proof of Theorem opeqpr
StepHypRef Expression
1 eqcom 2437 . 2
2 opeqpr.1 . . . 4
3 opeqpr.2 . . . 4
42, 3dfop 3975 . . 3
54eqeq2i 2445 . 2
6 opeqpr.3 . . 3
7 opeqpr.4 . . 3
8 snex 4397 . . 3
9 prex 4398 . . 3
106, 7, 8, 9preq12b 3966 . 2
111, 5, 103bitri 263 1
 Colors of variables: wff set class Syntax hints:   wb 177   wo 358   wa 359   wceq 1652   wcel 1725  cvv 2948  csn 3806  cpr 3807  cop 3809 This theorem is referenced by:  relop  5015 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  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-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815
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