Theorem frforeq2 4465

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 2740	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇)))
2	1	imbi1d 231	. . . 4 ⊢ (𝐴 = 𝐵 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
3	2	raleqbi1dv 2752	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
4		sseq1 3260	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝑇 ↔ 𝐵 ⊆ 𝑇))
5	3, 4	imbi12d 234	. 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 4451	. 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
7		df-frfor 4451	. 2 ⊢ ( FrFor 𝑅𝐵𝑇 ↔ (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐵 ⊆ 𝑇))
8	5, 6, 7	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520   ⊆ wss 3210   class class class wbr 4108   FrFor wfrfor 4447

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-in 3216  df-ss 3223  df-frfor 4451

This theorem is referenced by:  freq2  4466