Theorem frforeq3 4378

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

Assertion

Ref	Expression
frforeq3	⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))

Proof of Theorem frforeq3

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

Step	Hyp	Ref	Expression
1		eleq2 2257	. . . . . . 7 ⊢ (𝑆 = 𝑇 → (𝑦 ∈ 𝑆 ↔ 𝑦 ∈ 𝑇))
2	1	imbi2d 230	. . . . . 6 ⊢ (𝑆 = 𝑇 → ((𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) ↔ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇)))
3	2	ralbidv 2494	. . . . 5 ⊢ (𝑆 = 𝑇 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇)))
4		eleq2 2257	. . . . 5 ⊢ (𝑆 = 𝑇 → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ 𝑇))
5	3, 4	imbi12d 234	. . . 4 ⊢ (𝑆 = 𝑇 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
6	5	ralbidv 2494	. . 3 ⊢ (𝑆 = 𝑇 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
7		sseq2 3203	. . 3 ⊢ (𝑆 = 𝑇 → (𝐴 ⊆ 𝑆 ↔ 𝐴 ⊆ 𝑇))
8	6, 7	imbi12d 234	. 2 ⊢ (𝑆 = 𝑇 → ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇)))
9		df-frfor 4362	. 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆))
10		df-frfor 4362	. 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
11	8, 9, 10	3bitr4g 223	1 ⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3153   class class class wbr 4029   FrFor wfrfor 4358

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-in 3159  df-ss 3166  df-frfor 4362

This theorem is referenced by:  frind  4383