Theorem cbv3v 1768

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 1766 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbv3v.nf1	⊢ Ⅎ𝑦𝜑
cbv3v.nf2	⊢ Ⅎ𝑥𝜓
cbv3v.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbv3v	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbv3v

Step	Hyp	Ref	Expression
1		cbv3v.nf1	. 2 ⊢ Ⅎ𝑦𝜑
2		cbv3v.nf2	. 2 ⊢ Ⅎ𝑥𝜓
3		cbv3v.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 2, 3	cbv3 1766	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1371  Ⅎwnf 1484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  cbv1v  1771  cbvalv1  1775