Theorem cbv3 2414

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 36030. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker cbv3v 2354 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 2194	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	hbal 2173	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
4		cbv3.2	. . 3 ⊢ Ⅎ𝑥𝜓
5		cbv3.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
6	4, 5	spim 2404	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
7	3, 6	alrimih 1823	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1534  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 1969  ax-7 2014  ax-11 2160  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784

This theorem is referenced by:  cbval  2415  cbv1  2421  cbv3h  2423  axc16i  2457