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Theorem poeq12d 5601
Description: Equality deduction for partial orderings. (Contributed by Matthew House, 10-Sep-2025.)
Hypotheses
Ref Expression
poeq12d.1 (𝜑𝑅 = 𝑆)
poeq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
poeq12d (𝜑 → (𝑅 Po 𝐴𝑆 Po 𝐵))

Proof of Theorem poeq12d
StepHypRef Expression
1 poeq12d.1 . 2 (𝜑𝑅 = 𝑆)
2 poeq12d.2 . 2 (𝜑𝐴 = 𝐵)
3 poeq1 5599 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
4 poeq2 5600 . . 3 (𝐴 = 𝐵 → (𝑆 Po 𝐴𝑆 Po 𝐵))
53, 4sylan9bb 509 . 2 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑅 Po 𝐴𝑆 Po 𝐵))
61, 2, 5syl2anc 584 1 (𝜑 → (𝑅 Po 𝐴𝑆 Po 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536   Po wpo 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726  df-clel 2813  df-ral 3059  df-ss 3979  df-br 5148  df-po 5596
This theorem is referenced by:  weiunpo  36447
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