Theorem cbv2w 1738

Description: Rule used to change bound variables, using implicit substitution. Version of cbv2 1737 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbv2w.1	⊢ Ⅎ𝑥𝜑
cbv2w.2	⊢ Ⅎ𝑦𝜑
cbv2w.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbv2w.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbv2w.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbv2w	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem cbv2w

Step	Hyp	Ref	Expression
1		cbv2w.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		cbv2w.2	. . 3 ⊢ Ⅎ𝑦𝜑
3		cbv2w.3	. . 3 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		cbv2w.4	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbv2w.5	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6		biimp 117	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
7	5, 6	syl6 33	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
8	1, 2, 3, 4, 7	cbv1v 1735	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
9		equcomi 1692	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
10		biimpr 129	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓))
11	9, 5, 10	syl56 34	. . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → (𝜒 → 𝜓)))
12	2, 1, 4, 3, 11	cbv1v 1735	. 2 ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
13	8, 12	impbid 128	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  Ⅎwnf 1448

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-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by: (None)