Theorem a5i 1554

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 1551	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
2		ax-5 1458	. . 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 1462	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 1458  ax-gen 1460  ax-ial 1545

This theorem is referenced by:  hbae  1729  equveli  1770  hbsb2a  1817  hbsb2e  1818  aev  1823  dveeq2or  1827  hbsb2  1847  nfsb2or  1848  reu6  2941