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Theorem breq123d 3741
 Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
Hypotheses
Ref Expression
breq1d.1 (φA = B)
breq123d.2 (φ𝑅 = 𝑆)
breq123d.3 (φ𝐶 = 𝐷)
Assertion
Ref Expression
breq123d (φ → (A𝑅𝐶B𝑆𝐷))

Proof of Theorem breq123d
StepHypRef Expression
1 breq1d.1 . . 3 (φA = B)
2 breq123d.3 . . 3 (φ𝐶 = 𝐷)
31, 2breq12d 3740 . 2 (φ → (A𝑅𝐶B𝑅𝐷))
4 breq123d.2 . . 3 (φ𝑅 = 𝑆)
54breqd 3738 . 2 (φ → (B𝑅𝐷B𝑆𝐷))
63, 5bitrd 177 1 (φ → (A𝑅𝐶B𝑆𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1223   class class class wbr 3727 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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-un 2890  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728 This theorem is referenced by:  sbcbrg  3776  fmptco  5243
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