Theorem breqdi 5089

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 5085	. 2 ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷))
4	1, 3	mpbid 234	1 ⊢ (𝜑 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1548   class class class wbr 5074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-cleq 2733  df-clel 2816  df-br 5075

This theorem is referenced by:  rtrclreclem3  15017  episect  17747  dvef  25968  acopyeu  28922  isleagd  28936  weiunso  36707  0prjspn  43091  brfvimex  44483  brovmptimex  44484  ntrclsnvobr  44509  clsneibex  44559  neicvgbex  44569  up1st2nd  49687  up1st2ndr  49688