Theorem breq12 2067

Description: Equality theorem for binary relation.

Assertion

Ref	Expression
breq12	⊢ ((A = B ∧ C = D) → (ARC ↔ BRD))

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 2065	. 2 ⊢ (A = B → (ARC ↔ BRC))
2		breq2 2066	. 2 ⊢ (C = D → (BRC ↔ BRD))
3	1, 2	sylan9bb 418	1 ⊢ ((A = B ∧ C = D) → (ARC ↔ BRD))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   class class class wbr 2054

This theorem is referenced by:  breqan12d 2074  ersym 3209  canth2g 3382  zornlem6 3608  ltresr 4052

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063

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