Theorem frforeq2 4100

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

Assertion

Ref	Expression
frforeq2	⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))

Proof of Theorem frforeq2

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

Step	Hyp	Ref	Expression
1		raleq 2549	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇)))
2	1	imbi1d 229	. . . 4 ⊢ (𝐴 = 𝐵 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
3	2	raleqbi1dv 2557	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
4		sseq1 3020	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝑇 ↔ 𝐵 ⊆ 𝑇))
5	3, 4	imbi12d 232	. 2 ⊢ (𝐴 = 𝐵 → ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇) ↔ (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐵 ⊆ 𝑇)))
6		df-frfor 4086	. 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
7		df-frfor 4086	. 2 ⊢ ( FrFor 𝑅𝐵𝑇 ↔ (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐵 ⊆ 𝑇))
8	5, 6, 7	3bitr4g 221	1 ⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-in 2979  df-ss 2986  df-frfor 4086

This theorem is referenced by:  freq2  4101