Theorem seeq12d 5619

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 5617	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴))
4		seeq2 5618	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Se 𝐴 ↔ 𝑆 Se 𝐵))
5	3, 4	sylan9bb 517	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))
6	1, 2, 5	syl2anc 593	1 ⊢ (𝜑 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560   Se wse 5598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rab 3415  df-v 3456  df-in 3911  df-ss 3921  df-br 5101  df-se 5601

This theorem is referenced by: (None)