Theorem breqdi 5123

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 5119	. 2 ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → 𝐶𝐵𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   class class class wbr 5108

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-clel 2837  df-br 5109

This theorem is used by:  rtrclreclem3  15104  episect  17848  dvef  26150  acopyeu  29156  perpeqlem  29161  isleagd  29176  weiunso  37005  0prjspn  43388  brfvimex  44780  brovmptimex  44781  ntrclsnvobr  44806  clsneibex  44856  neicvgbex  44866  up1st2nd  49991  up1st2ndr  49992