Theorem cbv1v 1735

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)

Hypotheses

Ref	Expression
cbv1v.1	⊢ Ⅎ𝑥𝜑
cbv1v.2	⊢ Ⅎ𝑦𝜑
cbv1v.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbv1v.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbv1v.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
cbv1v	⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbv1v

Step	Hyp	Ref	Expression
1		cbv1v.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
2		cbv1v.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3	1, 2	nfim1 1559	. . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝜓)
4		cbv1v.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		cbv1v.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
6	4, 5	nfim1 1559	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒)
7		cbv1v.5	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
8	7	com12 30	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 → 𝜒)))
9	8	a2d 26	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
10	3, 6, 9	cbv3v 1732	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦(𝜑 → 𝜒))
11	4	19.21 1571	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12	1	19.21 1571	. . 3 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒))
13	10, 11, 12	3imtr3i 199	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒))
14	13	pm2.86i 98	1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Syntax hints:   → wi 4  ∀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-4 1498  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:  cbv2w  1738