Theorem breqd 4651

Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Hypothesis

Ref	Expression
breq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
breqd	⊢ (φ → (CAD ↔ CBD))

Proof of Theorem breqd

Step	Hyp	Ref	Expression
1		breq1d.1	. 2 ⊢ (φ → A = B)
2		breq 4642	. 2 ⊢ (A = B → (CAD ↔ CBD))
3	1, 2	syl 15	1 ⊢ (φ → (CAD ↔ CBD))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   class class class wbr 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-br 4641

This theorem is referenced by:  breq123d  4654  sbcbr12g  4687