Theorem cbv3h

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbv3hv 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.)

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

Proof

Step	Hyp	Ref	Expression
1		cbv3h.1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
2		cbv3h.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3		cbv3h.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
4	1	nf5i	⊢ Ⅎ 𝑦 𝜑
5	2	nf5i	⊢ Ⅎ 𝑥 𝜓
6	4 5 3	cbv3	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )