Theorem breqdi 5122

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 5118	. 2 ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1540   class class class wbr 5107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-clel 2803  df-br 5108

This theorem is referenced by:  rtrclreclem3  15026  episect  17747  dvef  25884  acopyeu  28761  isleagd  28775  weiunso  36454  0prjspn  42616  brfvimex  44015  brovmptimex  44016  ntrclsnvobr  44041  clsneibex  44091  neicvgbex  44101  up1st2nd  49174  up1st2ndr  49175