Theorem frforeq3 4229

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 2178	. . . . . . 7 ⊢ (𝑆 = 𝑇 → (𝑦 ∈ 𝑆 ↔ 𝑦 ∈ 𝑇))
2	1	imbi2d 229	. . . . . 6 ⊢ (𝑆 = 𝑇 → ((𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) ↔ (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇)))
3	2	ralbidv 2411	. . . . 5 ⊢ (𝑆 = 𝑇 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇)))
4		eleq2 2178	. . . . 5 ⊢ (𝑆 = 𝑇 → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ 𝑇))
5	3, 4	imbi12d 233	. . . 4 ⊢ (𝑆 = 𝑇 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
6	5	ralbidv 2411	. . 3 ⊢ (𝑆 = 𝑇 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
7		sseq2 3087	. . 3 ⊢ (𝑆 = 𝑇 → (𝐴 ⊆ 𝑆 ↔ 𝐴 ⊆ 𝑇))
8	6, 7	imbi12d 233	. 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 4213	. 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆))
10		df-frfor 4213	. 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
11	8, 9, 10	3bitr4g 222	1 ⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1314   ∈ wcel 1463  ∀wral 2390   ⊆ wss 3037   class class class wbr 3895   FrFor wfrfor 4209

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2395  df-in 3043  df-ss 3050  df-frfor 4213

This theorem is referenced by:  frind  4234