Theorem tapeq1 7471

Description: Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)

Assertion

Ref	Expression
tapeq1	⊢ (𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))

Proof of Theorem tapeq1

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

Step	Hyp	Ref	Expression
1		sseq1 3250	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 ⊆ (𝐴 × 𝐴) ↔ 𝑆 ⊆ (𝐴 × 𝐴)))
2		breq 4090	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑥 ↔ 𝑥𝑆𝑥))
3	2	notbid 673	. . . . 5 ⊢ (𝑅 = 𝑆 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑆𝑥))
4	3	ralbidv 2532	. . . 4 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑆𝑥))
5		breq 4090	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦))
6		breq 4090	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥))
7	5, 6	imbi12d 234	. . . . 5 ⊢ (𝑅 = 𝑆 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (𝑥𝑆𝑦 → 𝑦𝑆𝑥)))
8	7	2ralbidv 2556	. . . 4 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 → 𝑦𝑆𝑥)))
9	4, 8	anbi12d 473	. . 3 ⊢ (𝑅 = 𝑆 → ((∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑆𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 → 𝑦𝑆𝑥))))
10		breq 4090	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑧 ↔ 𝑥𝑆𝑧))
11		breq 4090	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧))
12	10, 11	orbi12d 800	. . . . . . 7 ⊢ (𝑅 = 𝑆 → ((𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧) ↔ (𝑥𝑆𝑧 ∨ 𝑦𝑆𝑧)))
13	5, 12	imbi12d 234	. . . . . 6 ⊢ (𝑅 = 𝑆 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑦𝑆𝑧))))
14	13	ralbidv 2532	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑦𝑆𝑧))))
15	14	2ralbidv 2556	. . . 4 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑦𝑆𝑧))))
16	5	notbid 673	. . . . . 6 ⊢ (𝑅 = 𝑆 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑆𝑦))
17	16	imbi1d 231	. . . . 5 ⊢ (𝑅 = 𝑆 → ((¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦) ↔ (¬ 𝑥𝑆𝑦 → 𝑥 = 𝑦)))
18	17	2ralbidv 2556	. . . 4 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑆𝑦 → 𝑥 = 𝑦)))
19	15, 18	anbi12d 473	. . 3 ⊢ (𝑅 = 𝑆 → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑦𝑆𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑆𝑦 → 𝑥 = 𝑦))))
20	1, 9, 19	3anbi123d 1348	. 2 ⊢ (𝑅 = 𝑆 → ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))) ↔ (𝑆 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑆𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 → 𝑦𝑆𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑦𝑆𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑆𝑦 → 𝑥 = 𝑦)))))
21		dftap2 7470	. 2 ⊢ (𝑅 TAp 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))))
22		dftap2 7470	. 2 ⊢ (𝑆 TAp 𝐴 ↔ (𝑆 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑆𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑆𝑦 → 𝑦𝑆𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑆𝑦 → (𝑥𝑆𝑧 ∨ 𝑦𝑆𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑆𝑦 → 𝑥 = 𝑦))))
23	20, 21, 22	3bitr4g 223	1 ⊢ (𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   ∧ w3a 1004   = wceq 1397  ∀wral 2510   ⊆ wss 3200   class class class wbr 4088   × cxp 4723   TAp wtap 7468

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-in 3206  df-ss 3213  df-br 4089  df-pap 7467  df-tap 7469

This theorem is referenced by:  2omotaplemst  7477  exmidapne  7479  exmidmotap  7480