Theorem cbv3 2265

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 34175. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)

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	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfal 2153	. 2 ⊢ Ⅎ𝑦∀𝑥𝜑
3		cbv3.2	. . 3 ⊢ Ⅎ𝑥𝜓
4		cbv3.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	spim 2254	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
6	2, 5	alrimi 2082	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  cbv3h  2266  cbv1  2267  cbval  2271  axc16i  2322  bj-mo3OLD  32832