Theorem cbv2h 2421

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-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		biimp 207	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
5	3, 4	syl6 35	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
6	1, 2, 5	cbv1h 2420	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
7		equcomi 2121	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
8		biimpr 212	. . . . 5 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓))
9	7, 3, 8	syl56 36	. . . 4 ⊢ (𝜑 → (𝑦 = 𝑥 → (𝜒 → 𝜓)))
10	2, 1, 9	cbv1h 2420	. . 3 ⊢ (∀𝑦∀𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
11	10	alcoms 2208	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
12	6, 11	impbid 204	1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883

This theorem is referenced by:  cbv2  2422  eujustALT  2643