Theorem cbv1h 1746

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)

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 1541	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑
2		nfa2 1579	. 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦𝜑
3		sp 1511	. . . . 5 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	sps 1537	. . . 4 ⊢ (∀𝑥∀𝑦𝜑 → 𝜑)
5		cbv1h.1	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
6	4, 5	syl 14	. . 3 ⊢ (∀𝑥∀𝑦𝜑 → (𝜓 → ∀𝑦𝜓))
7	2, 6	nfd 1523	. 2 ⊢ (∀𝑥∀𝑦𝜑 → Ⅎ𝑦𝜓)
8		cbv1h.2	. . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
9	4, 8	syl 14	. . 3 ⊢ (∀𝑥∀𝑦𝜑 → (𝜒 → ∀𝑥𝜒))
10	1, 9	nfd 1523	. 2 ⊢ (∀𝑥∀𝑦𝜑 → Ⅎ𝑥𝜒)
11		cbv1h.3	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
12	4, 11	syl 14	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
13	1, 2, 7, 10, 12	cbv1 1745	1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Syntax hints:   → wi 4  ∀wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  cbv2h  1748