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Metamath Proof Explorer

Theorem hbal

Description: If x is not free in ph , it is not free in A. y ph . (Contributed by NM, 12-Mar-1993)

Ref Expression
Hypothesis hbal.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
Assertion hbal ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )

Proof

Step Hyp Ref Expression
1 hbal.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2 1 alimi ⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
3 ax-11 ⊢ ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )
4 2 3 syl ⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )