Theorem breqdi 4819

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 4815	. 2 ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷))
4	1, 3	mpbid 222	1 ⊢ (𝜑 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1632   class class class wbr 4804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2753  df-clel 2756  df-br 4805

This theorem is referenced by:  brfvimex  38826  brovmptimex  38827  ntrclsnvobr  38852  clsneibex  38902  neicvgbex  38912