Theorem breq1dd 32630

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 5106	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐵𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1541   class class class wbr 5096

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 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097

This theorem is referenced by:  mplvrpmfgalem  33658