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 ) )