Theorem a5i 1557

Description: Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
a5i.1	⊢ (∀𝑥𝜑 → 𝜓)

Assertion

Ref	Expression
a5i	⊢ (∀𝑥𝜑 → ∀𝑥𝜓)

Proof of Theorem a5i

Step	Hyp	Ref	Expression
1		hba1 1554	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
2		ax-5 1461	. . 3 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓))
3	1, 2	syl5 32	. 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
4		a5i.1	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5	3, 4	mpg 1465	1 ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1461  ax-gen 1463  ax-ial 1548

This theorem is referenced by:  hbae  1732  equveli  1773  hbsb2a  1820  hbsb2e  1821  aev  1826  dveeq2or  1830  hbsb2  1850  nfsb2or  1851  reu6  2953