Theorem soeq12d 5620

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 5618	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))
4		soeq2 5619	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Or 𝐴 ↔ 𝑆 Or 𝐵))
5	3, 4	sylan9bb 509	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
6	1, 2, 5	syl2anc 584	1 ⊢ (𝜑 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   Or wor 5596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-ex 1777  df-cleq 2727  df-clel 2814  df-ral 3060  df-ss 3980  df-br 5149  df-po 5597  df-so 5598

This theorem is referenced by:  opsrtoslem2  22098  weiunso  36449