Theorem breqdi 5093

Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)

Hypotheses

Ref	Expression
breq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
breqdi.1	⊢ (𝜑 → 𝐶𝐴𝐷)

Assertion

Ref	Expression
breqdi	⊢ (𝜑 → 𝐶𝐵𝐷)

Proof of Theorem breqdi

Step	Hyp	Ref	Expression
1		breqdi.1	. 2 ⊢ (𝜑 → 𝐶𝐴𝐷)
2		breq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	breqd 5089	. 2 ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷))
4	1, 3	mpbid 231	1 ⊢ (𝜑 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1541   class class class wbr 5078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-cleq 2731  df-clel 2817  df-br 5079

This theorem is referenced by:  rtrclreclem3  14752  episect  17478  dvef  25125  acopyeu  27176  isleagd  27190  0prjspn  40445  brfvimex  41589  brovmptimex  41590  ntrclsnvobr  41615  clsneibex  41665  neicvgbex  41675