Theorem breq1dd 5124

Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 10-Jan-2026.)

Hypotheses

Ref	Expression
breq1dd.1	⊢ (𝜑 → 𝐴 = 𝐵)
breq1dd.2	⊢ (𝜑 → 𝐴𝑅𝐶)

Assertion

Ref	Expression
breq1dd	⊢ (𝜑 → 𝐵𝑅𝐶)

Proof of Theorem breq1dd

Step	Hyp	Ref	Expression
1		breq1dd.2	. 2 ⊢ (𝜑 → 𝐴𝑅𝐶)
2		breq1dd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	breq1d 5116	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → 𝐵𝑅𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   class class class wbr 5106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107

This theorem is used by:  perpeq  29363  mplvrpmfgalem  34232  fucoaltfunc  50633