Theorem breqan12rd 3891

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Hypotheses

Ref	Expression
breq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
breqan12i.2	⊢ (𝜓 → 𝐶 = 𝐷)

Assertion

Ref	Expression
breqan12rd	⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))

Proof of Theorem breqan12rd

Step	Hyp	Ref	Expression
1		breq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		breqan12i.2	. . 3 ⊢ (𝜓 → 𝐶 = 𝐷)
3	1, 2	breqan12d 3890	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	3	ancoms 266	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1299   class class class wbr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876

This theorem is referenced by:  xltnegi  9459