Theorem a5i 1417

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 1415	. . 3 ⊢ (∀xφ → ∀x∀xφ)
2		ax-5 1316	. . 3 ⊢ (∀x(∀xφ → ψ) → (∀x∀xφ → ∀xψ))
3	1, 2	syl5 28	. 2 ⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))
4		a5i.1	. 2 ⊢ (∀xφ → ψ)
5	3, 4	mpg 1320	1 ⊢ (∀xφ → ∀xψ)

Syntax hints:   → wi 4  ∀wal 1226

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-5 1316  ax-gen 1318  ax-ial 1409

This theorem is referenced by:  hbae  1588  equveli  1624  hbsb2a  1669  hbsb2e  1670  aev  1675  dveeq2or  1679  hbsb2  1699  nfsb2or  1700  reu6  2707