Theorem cbv3h 2424

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbv3hv 2360 if possible. (Contributed by NM, 8-Jun-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)

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 2150	. 2 ⊢ Ⅎ𝑦𝜑
3		cbv3h.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2150	. 2 ⊢ Ⅎ𝑥𝜓
5		cbv3h.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
6	2, 4, 5	cbv3 2415	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)