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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 → ( 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 ) )