Theorem cbv3h 1678

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)

Hypotheses

Ref	Expression
cbv3h.1	⊢ (𝜑 → ∀𝑦𝜑)
cbv3h.2	⊢ (𝜓 → ∀𝑥𝜓)
cbv3h.3	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbv3h	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Proof of Theorem cbv3h

Step	Hyp	Ref	Expression
1		cbv3h.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	nfi 1396	. 2 ⊢ Ⅎ𝑦𝜑
3		cbv3h.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nfi 1396	. 2 ⊢ Ⅎ𝑥𝜓
5		cbv3h.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
6	2, 4, 5	cbv3 1677	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395

This theorem is referenced by:  cbvalh  1683  ax16  1741  ax16i  1786  cleqh  2187