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Metamath Proof Explorer

Theorem cbv1v

Description: Rule used to change bound variables, using implicit substitution. Version of cbv1 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 16-Jun-2019)

Ref Expression
Hypotheses cbv1v.1 ⊢ Ⅎ 𝑥 𝜑
cbv1v.2 ⊢ Ⅎ 𝑦 𝜑
cbv1v.3 ⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
cbv1v.4 ⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
cbv1v.5 ⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
Assertion cbv1v ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )

Proof

Step Hyp Ref Expression
1 cbv1v.1 ⊢ Ⅎ 𝑥 𝜑
2 cbv1v.2 ⊢ Ⅎ 𝑦 𝜑
3 cbv1v.3 ⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
4 cbv1v.4 ⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
5 cbv1v.5 ⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
6 2 3 nfim1 ⊢ Ⅎ 𝑦 ( 𝜑 → 𝜓 )
7 1 4 nfim1 ⊢ Ⅎ 𝑥 ( 𝜑 → 𝜒 )
8 5 com12 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
9 8 a2d ⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
10 6 7 9 cbv3v ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∀ 𝑦 ( 𝜑 → 𝜒 ) )
11 1 19.21 ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) )
12 2 19.21 ⊢ ( ∀ 𝑦 ( 𝜑 → 𝜒 ) ↔ ( 𝜑 → ∀ 𝑦 𝜒 ) )
13 10 11 12 3imtr3i ⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ( 𝜑 → ∀ 𝑦 𝜒 ) )
14 13 pm2.86i ⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )