Theorem breqdi 5104

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 5100	. 2 ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1541   class class class wbr 5089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-clel 2806  df-br 5090

This theorem is referenced by:  rtrclreclem3  14967  episect  17692  dvef  25911  acopyeu  28812  isleagd  28826  weiunso  36510  0prjspn  42720  brfvimex  44118  brovmptimex  44119  ntrclsnvobr  44144  clsneibex  44194  neicvgbex  44204  up1st2nd  49285  up1st2ndr  49286