Metamath Proof Explorer


Theorem bj-cbv1hv

Description: Version of cbv1h with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

Ref Expression
Hypotheses bj-cbv1hv.1 ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
bj-cbv1hv.2 ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
bj-cbv1hv.3 ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓𝜒 ) ) )
Assertion bj-cbv1hv ( ∀ 𝑥𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )

Proof

Step Hyp Ref Expression
1 bj-cbv1hv.1 ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
2 bj-cbv1hv.2 ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
3 bj-cbv1hv.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 cbv1v ( ∀ 𝑥𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )