Theorem cbv1 1722

Description: Rule used to change bound variables, using implicit substitution. Revised to format 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
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 1550	. . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝜓)
4		cbv1.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		cbv1.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
6	4, 5	nfim1 1550	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒)
7		cbv1.5	. . . . . 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	cbv3 1720	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦(𝜑 → 𝜒))
11	4	19.21 1562	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12	1	19.21 1562	. . 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 1329  Ⅎwnf 1436

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  cbv1h  1723