Theorem cbv1h

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv1h.1	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
	cbv1h.2	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
	cbv1h.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
Assertion	cbv1h	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbv1h.1	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
2		cbv1h.2	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
3		cbv1h.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
4		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ∀ 𝑦 𝜑
5		nfa2	⊢ Ⅎ 𝑦 ∀ 𝑥 ∀ 𝑦 𝜑
6		2sp	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜑 )
7	6 1	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
8	5 7	nf5d	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → Ⅎ 𝑦 𝜓 )
9	6 2	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
10	4 9	nf5d	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → Ⅎ 𝑥 𝜒 )
11	6 3	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
12	4 5 8 10 11	cbv1	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )