Theorem cbv3 1753

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 1755 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 1587	. 2 ⊢ Ⅎ𝑦∀𝑥𝜑
3		cbv3.2	. . 3 ⊢ Ⅎ𝑥𝜓
4		cbv3.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	spim 1749	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
6	2, 5	alrimi 1533	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1362  Ⅎwnf 1471

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  cbv3h  1754  cbv3v  1755  cbv1  1756  mo2n  2070  mo23  2083  setindis  15459  bdsetindis  15461