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Theorem eqbrriv 4358
Description: Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Hypotheses
Ref Expression
eqbrriv.1 Rel A
eqbrriv.2 Rel B
eqbrriv.3 (xAyxBy)
Assertion
Ref Expression
eqbrriv A = B
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem eqbrriv
StepHypRef Expression
1 eqbrriv.1 . 2 Rel A
2 eqbrriv.2 . 2 Rel B
3 eqbrriv.3 . . 3 (xAyxBy)
4 df-br 3735 . . 3 (xAy ↔ ⟨x, y A)
5 df-br 3735 . . 3 (xBy ↔ ⟨x, y B)
63, 4, 53bitr3i 199 . 2 (⟨x, y A ↔ ⟨x, y B)
71, 2, 6eqrelriiv 4357 1 A = B
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1226   wcel 1370  cop 3349   class class class wbr 3734  Rel wrel 4273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275
This theorem is referenced by:  resco  4748  tpostpos  5797
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