Theorem papeq2 7604

Description: Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)

Assertion

Ref	Expression
papeq2	⊢ (𝐴 = 𝐵 → (𝑅 Ap 𝐴 ↔ 𝑅 Ap 𝐵))

Proof of Theorem papeq2

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

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
2	1	sqxpeqd 4798	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
3	2	sseq2d 3278	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ (𝐴 × 𝐴) ↔ 𝑅 ⊆ (𝐵 × 𝐵)))
4		raleq 2749	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑥))
5	3, 4	anbi12d 477	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ↔ (𝑅 ⊆ (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑥)))
6		raleq 2749	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
7	6	raleqbi1dv 2761	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
8		raleq 2749	. . . . . 6 ⊢ (𝐴 = 𝐵 → (∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧))))
9	8	raleqbi1dv 2761	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧))))
10	9	raleqbi1dv 2761	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧))))
11	7, 10	anbi12d 477	. . 3 ⊢ (𝐴 = 𝐵 → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧))) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
12	5, 11	anbi12d 477	. 2 ⊢ (𝐴 = 𝐵 → (((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))) ↔ ((𝑅 ⊆ (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧))))))
13		df-pap 7602	. 2 ⊢ (𝑅 Ap 𝐴 ↔ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
14		df-pap 7602	. 2 ⊢ (𝑅 Ap 𝐵 ↔ ((𝑅 ⊆ (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
15	12, 13, 14	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝑅 Ap 𝐴 ↔ 𝑅 Ap 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ∀wral 2528   ⊆ wss 3220   class class class wbr 4128   × cxp 4770   Ap wap 7601

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-in 3226  df-ss 3233  df-opab 4191  df-xp 4778  df-pap 7602

This theorem is referenced by:  opprdrng  14603