Theorem frforeq1 4098

Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)

Assertion

Ref	Expression
frforeq1	⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))

Proof of Theorem frforeq1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 3787	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥))
2	1	imbi1d 229	. . . . . 6 ⊢ (𝑅 = 𝑆 → ((𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇)))
3	2	ralbidv 2368	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇)))
4	3	imbi1d 229	. . . 4 ⊢ (𝑅 = 𝑆 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
5	4	ralbidv 2368	. . 3 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
6	5	imbi1d 229	. 2 ⊢ (𝑅 = 𝑆 → ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇)))
7		df-frfor 4086	. 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
8		df-frfor 4086	. 2 ⊢ ( FrFor 𝑆𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
9	6, 7, 8	3bitr4g 221	1 ⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348   ⊆ wss 2973   class class class wbr 3785   FrFor wfrfor 4082

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077  df-ral 2353  df-br 3786  df-frfor 4086

This theorem is referenced by:  freq1  4099