Metamath Proof Explorer


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 → ( 𝜓𝜒 ) ) )
Assertion cbv1h ( ∀ 𝑥𝑦 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 → ( 𝜓𝜒 ) ) )
4 nfa1 𝑥𝑥𝑦 𝜑
5 nfa2 𝑦𝑥𝑦 𝜑
6 2sp ( ∀ 𝑥𝑦 𝜑𝜑 )
7 6 1 syl ( ∀ 𝑥𝑦 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
8 5 7 nf5d ( ∀ 𝑥𝑦 𝜑 → Ⅎ 𝑦 𝜓 )
9 6 2 syl ( ∀ 𝑥𝑦 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
10 4 9 nf5d ( ∀ 𝑥𝑦 𝜑 → Ⅎ 𝑥 𝜒 )
11 6 3 syl ( ∀ 𝑥𝑦 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓𝜒 ) ) )
12 4 5 8 10 11 cbv1 ( ∀ 𝑥𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )