Theorem cbv2h 1707

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

Hypotheses

Ref	Expression
cbv2h.1	⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
cbv2h.2	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
cbv2h.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbv2h	⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2h

Step	Hyp	Ref	Expression
1		cbv2h.1	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
2		cbv2h.2	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
3		cbv2h.3	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
4		bi1 117	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
5	3, 4	syl6 33	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
6	1, 2, 5	cbv1h 1706	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
7		equcomi 1663	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
8		bi2 129	. . . . 5 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓))
9	7, 3, 8	syl56 34	. . . 4 ⊢ (𝜑 → (𝑦 = 𝑥 → (𝜒 → 𝜓)))
10	2, 1, 9	cbv1h 1706	. . 3 ⊢ (∀𝑦∀𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
11	10	a7s 1413	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
12	6, 11	impbid 128	1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1312

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498

This theorem depends on definitions:  df-bi 116  df-nf 1420

This theorem is referenced by:  cbv2  1708