Theorem frforeq2 4323

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 2661	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) ↔ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇)))
2	1	imbi1d 230	. . . 4 ⊢ (𝐴 = 𝐵 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
3	2	raleqbi1dv 2669	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
4		sseq1 3165	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝑇 ↔ 𝐵 ⊆ 𝑇))
5	3, 4	imbi12d 233	. 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 4309	. 2 ⊢ ( FrFor 𝑅𝐴𝑇 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐴 ⊆ 𝑇))
7		df-frfor 4309	. 2 ⊢ ( FrFor 𝑅𝐵𝑇 ↔ (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇) → 𝐵 ⊆ 𝑇))
8	5, 6, 7	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116   class class class wbr 3982   FrFor wfrfor 4305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-in 3122  df-ss 3129  df-frfor 4309

This theorem is referenced by:  freq2  4324