Theorem frforeq1 4433

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 4084	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥))
2	1	imbi1d 231	. . . . . 6 ⊢ (𝑅 = 𝑆 → ((𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇)))
3	2	ralbidv 2530	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇)))
4	3	imbi1d 231	. . . 4 ⊢ (𝑅 = 𝑆 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
5	4	ralbidv 2530	. . 3 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
6	5	imbi1d 231	. 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 4421	. 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
8		df-frfor 4421	. 2 ⊢ ( FrFor 𝑆𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑆𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
9	6, 7, 8	3bitr4g 223	1 ⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197   class class class wbr 4082   FrFor wfrfor 4417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-cleq 2222  df-clel 2225  df-ral 2513  df-br 4083  df-frfor 4421

This theorem is referenced by:  freq1  4434