Theorem breq1dd 5131

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 5123	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → 𝐵𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1567   class class class wbr 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114

This theorem is referenced by:  perpeq  29106  mplvrpmfgalem  33879