Theorem cbv1h 2410

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbv1h.1	⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
cbv1h.2	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
cbv1h.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
cbv1h	⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Proof of Theorem cbv1h

Step	Hyp	Ref	Expression
1		nfa1 2152	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
2		nfa2 2177	. 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦𝜑
3		2sp 2187	. . . 4 ⊢ (∀𝑥∀𝑦𝜑 → 𝜑)
4		cbv1h.1	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
5	3, 4	syl 17	. . 3 ⊢ (∀𝑥∀𝑦𝜑 → (𝜓 → ∀𝑦𝜓))
6	2, 5	nf5d 2285	. 2 ⊢ (∀𝑥∀𝑦𝜑 → Ⅎ𝑦𝜓)
7		cbv1h.2	. . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
8	3, 7	syl 17	. . 3 ⊢ (∀𝑥∀𝑦𝜑 → (𝜒 → ∀𝑥𝜒))
9	1, 8	nf5d 2285	. 2 ⊢ (∀𝑥∀𝑦𝜑 → Ⅎ𝑥𝜒)
10		cbv1h.3	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
11	3, 10	syl 17	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
12	1, 2, 6, 9, 11	cbv1 2407	1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Syntax hints:   → wi 4  ∀wal 1538

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 2008  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784

This theorem is referenced by:  cbv2h  2411