Theorem cbv3h 2266

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 8-Jun-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	nf5i 2024	. 2 ⊢ Ⅎ𝑦𝜑
3		cbv3h.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2024	. 2 ⊢ Ⅎ𝑥𝜓
5		cbv3h.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
6	2, 4, 5	cbv3 2265	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1481

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: (None)