Theorem breq1dd 32569

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 5105	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐵𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1540   class class class wbr 5095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096

This theorem is referenced by:  mplvrpmfgalem  33564