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Theorem seeq12d 5657
Description: Equality deduction for the set-like predicate. (Contributed by Matthew House, 10-Sep-2025.)
Hypotheses
Ref Expression
seeq12d.1 (𝜑𝑅 = 𝑆)
seeq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
seeq12d (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))

Proof of Theorem seeq12d
StepHypRef Expression
1 seeq12d.1 . 2 (𝜑𝑅 = 𝑆)
2 seeq12d.2 . 2 (𝜑𝐴 = 𝐵)
3 seeq1 5655 . . 3 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
4 seeq2 5656 . . 3 (𝐴 = 𝐵 → (𝑆 Se 𝐴𝑆 Se 𝐵))
53, 4sylan9bb 509 . 2 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑅 Se 𝐴𝑆 Se 𝐵))
61, 2, 5syl2anc 584 1 (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540   Se wse 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-br 5144  df-se 5638
This theorem is referenced by: (None)
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