Theorem cbv3 1756

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because proofs are encouraged to use the weaker cbv3v 1758 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	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfal 1590	. 2 ⊢ Ⅎ𝑦∀𝑥𝜑
3		cbv3.2	. . 3 ⊢ Ⅎ𝑥𝜓
4		cbv3.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	spim 1752	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
6	2, 5	alrimi 1536	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1362  Ⅎwnf 1474

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  cbv3h  1757  cbv3v  1758  cbv1  1759  mo2n  2073  mo23  2086  setindis  15613  bdsetindis  15615