Theorem cbv3 2402

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 38891. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker cbv3v 2337 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbv3.1	⊢ Ⅎ𝑦𝜑
cbv3.2	⊢ Ⅎ𝑥𝜓
cbv3.3	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbv3	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Proof of Theorem cbv3

Step	Hyp	Ref	Expression
1		cbv3.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	nf5ri 2195	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	hbal 2167	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
4		cbv3.2	. . 3 ⊢ Ⅎ𝑥𝜓
5		cbv3.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
6	4, 5	spim 2392	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
7	3, 6	alrimih 1824	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1538  Ⅎwnf 1783

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 2007  ax-11 2157  ax-12 2177  ax-13 2377

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

This theorem is referenced by:  cbval  2403  cbv1  2407  cbv3h  2409  axc16i  2441