Theorem seeq12d 5583

Description: Equality deduction for the set-like predicate. (Contributed by Matthew House, 10-Sep-2025.)

Hypotheses

Ref	Expression
seeq12d.1	⊢ (𝜑 → 𝑅 = 𝑆)
seeq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
seeq12d	⊢ (𝜑 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Proof of Theorem seeq12d

Step	Hyp	Ref	Expression
1		seeq12d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		seeq12d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		seeq1 5581	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴))
4		seeq2 5582	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Se 𝐴 ↔ 𝑆 Se 𝐵))
5	3, 4	sylan9bb 509	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))
6	1, 2, 5	syl2anc 584	1 ⊢ (𝜑 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   Se wse 5562

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rab 3396  df-v 3438  df-in 3904  df-ss 3914  df-br 5087  df-se 5565

This theorem is referenced by: (None)