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Metamath Proof Explorer

Theorem hbald

Description: Deduction form of bound-variable hypothesis builder hbal . (Contributed by NM, 2-Jan-2002)

Ref Expression
Hypotheses hbald.1 ⊢ φ → ∀ y φ
hbald.2 ⊢ φ → ψ → ∀ x ψ
Assertion hbald ⊢ φ → ∀ y ψ → ∀ x ∀ y ψ

Proof

Step Hyp Ref Expression
1 hbald.1 ⊢ φ → ∀ y φ
2 hbald.2 ⊢ φ → ψ → ∀ x ψ
3 1 2 alimdh ⊢ φ → ∀ y ψ → ∀ y ∀ x ψ
4 ax-11 ⊢ ∀ y ∀ x ψ → ∀ x ∀ y ψ
5 3 4 syl6 ⊢ φ → ∀ y ψ → ∀ x ∀ y ψ