Theorem soeq12d 5583

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 5581	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))
4		soeq2 5582	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Or 𝐴 ↔ 𝑆 Or 𝐵))
5	3, 4	sylan9bb 518	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
6	1, 2, 5	syl2anc 595	1 ⊢ (𝜑 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563   Or wor 5559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-ex 1803  df-cleq 2757  df-clel 2840  df-ral 3080  df-ss 3924  df-br 5106  df-po 5560  df-so 5561

This theorem is referenced by:  opsrtoslem2  22167  weiunso  36839