Theorem seeq12d 5672

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 5670	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴))
4		seeq2 5671	. . 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 583	1 ⊢ (𝜑 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   Se wse 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-br 5167  df-se 5653

This theorem is referenced by: (None)