Theorem cbv2 1635

Description: Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)

Hypotheses

Ref	Expression
cbv2.1	⊢ Ⅎ𝑥𝜑
cbv2.2	⊢ Ⅎ𝑦𝜑
cbv2.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbv2.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbv2.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbv2	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2

Step	Hyp	Ref	Expression
1		cbv2.2	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1412	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbv2.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
4	3	nfal 1468	. . . 4 ⊢ Ⅎ𝑥∀𝑦𝜑
5	4	nfri 1412	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
6	2, 5	syl 14	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜑)
7		cbv2.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝜓)
8	7	nfrd 1413	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
9		cbv2.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
10	9	nfrd 1413	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
11		cbv2.5	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
12	8, 10, 11	cbv2h 1634	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
13	6, 12	syl 14	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  cbvald  1800  cbvrald  9927