Theorem soeq12d 5576

Description: Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.) (Proof shortened by Matthew House, 10-Sep-2025.)

Hypotheses

Ref	Expression
soeq12d.1	⊢ (𝜑 → 𝑅 = 𝑆)
soeq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
soeq12d	⊢ (𝜑 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))

Proof of Theorem soeq12d

Step	Hyp	Ref	Expression
1		soeq12d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		soeq12d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		soeq1 5574	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))
4		soeq2 5575	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Or 𝐴 ↔ 𝑆 Or 𝐵))
5	3, 4	sylan9bb 517	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
6	1, 2, 5	syl2anc 593	1 ⊢ (𝜑 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   Or wor 5552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-ex 1799  df-cleq 2753  df-clel 2836  df-ral 3076  df-ss 3921  df-br 5100  df-po 5553  df-so 5554

This theorem is referenced by:  opsrtoslem2  22089  weiunso  36790