Theorem poeq12d 5527

Description: Equality deduction for partial orderings. (Contributed by Matthew House, 10-Sep-2025.)

Hypotheses

Ref	Expression
poeq12d.1	⊢ (𝜑 → 𝑅 = 𝑆)
poeq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
poeq12d	⊢ (𝜑 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵))

Proof of Theorem poeq12d

Step	Hyp	Ref	Expression
1		poeq12d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		poeq12d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		poeq1 5525	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴))
4		poeq2 5526	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Po 𝐴 ↔ 𝑆 Po 𝐵))
5	3, 4	sylan9bb 509	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵))
6	1, 2, 5	syl2anc 584	1 ⊢ (𝜑 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   Po wpo 5520

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 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2722  df-clel 2804  df-ral 3046  df-ss 3917  df-br 5090  df-po 5522

This theorem is referenced by:  weiunpo  36478