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Theorem eqrelriv 4376
Description: Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
Hypothesis
Ref Expression
eqrelriv.1 (⟨x, y A ↔ ⟨x, y B)
Assertion
Ref Expression
eqrelriv ((Rel A Rel B) → A = B)
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem eqrelriv
StepHypRef Expression
1 eqrelriv.1 . . 3 (⟨x, y A ↔ ⟨x, y B)
21gen2 1336 . 2 xy(⟨x, y A ↔ ⟨x, y B)
3 eqrel 4372 . 2 ((Rel A Rel B) → (A = Bxy(⟨x, y A ↔ ⟨x, y B)))
42, 3mpbiri 157 1 ((Rel A Rel B) → A = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  cop 3370  Rel wrel 4293
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by:  eqrelriiv  4377  dfrel2  4714  coi1  4779  cnviinm  4802
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