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Theorem eqbrrdv 4360
Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
eqbrrdv.1 (φ → Rel A)
eqbrrdv.2 (φ → Rel B)
eqbrrdv.3 (φ → (xAyxBy))
Assertion
Ref Expression
eqbrrdv (φA = B)
Distinct variable groups:   x,y,A   x,B,y   φ,x,y

Proof of Theorem eqbrrdv
StepHypRef Expression
1 eqbrrdv.3 . . . 4 (φ → (xAyxBy))
2 df-br 3735 . . . 4 (xAy ↔ ⟨x, y A)
3 df-br 3735 . . . 4 (xBy ↔ ⟨x, y B)
41, 2, 33bitr3g 211 . . 3 (φ → (⟨x, y A ↔ ⟨x, y B))
54alrimivv 1733 . 2 (φxy(⟨x, y A ↔ ⟨x, y B))
6 eqbrrdv.1 . . 3 (φ → Rel A)
7 eqbrrdv.2 . . 3 (φ → Rel B)
8 eqrel 4352 . . 3 ((Rel A Rel B) → (A = Bxy(⟨x, y A ↔ ⟨x, y B)))
96, 7, 8syl2anc 393 . 2 (φ → (A = Bxy(⟨x, y A ↔ ⟨x, y B)))
105, 9mpbird 156 1 (φA = B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1224   = 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:  eqbrrdva  4428
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