Theorem cbv1 2407

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. See cbv1v 2338 with disjoint variable conditions, not depending on ax-13 2377. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbv1.1	⊢ Ⅎ𝑥𝜑
cbv1.2	⊢ Ⅎ𝑦𝜑
cbv1.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbv1.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbv1.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
cbv1	⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Proof of Theorem cbv1

Step	Hyp	Ref	Expression
1		cbv1.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
2		cbv1.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3	1, 2	nfim1 2199	. . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝜓)
4		cbv1.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		cbv1.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
6	4, 5	nfim1 2199	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒)
7		cbv1.5	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
8	7	com12 32	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 → 𝜒)))
9	8	a2d 29	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
10	3, 6, 9	cbv3 2402	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦(𝜑 → 𝜒))
11	4	19.21 2207	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12	1	19.21 2207	. . 3 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒))
13	10, 11, 12	3imtr3i 291	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒))
14	13	pm2.86i 110	1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Syntax hints:   → wi 4  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-11 2157  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784

This theorem is referenced by:  cbv2  2408  cbv1h  2410