Theorem sylg 1825

Description: A syllogism combined with generalization. Inference associated with sylgt 1824. General form of alrimih 1826. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1826. (Revised by BJ, 4-Oct-2019.)

Hypotheses

Ref	Expression
sylg.1	⊢ (𝜑 → ∀𝑥𝜓)
sylg.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
sylg	⊢ (𝜑 → ∀𝑥𝜒)

Proof of Theorem sylg

Step	Hyp	Ref	Expression
1		sylg.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
2		sylg.2	. . 3 ⊢ (𝜓 → 𝜒)
3	2	alimi 1813	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1797  ax-4 1811

This theorem is referenced by:  alrimih  1826  ax9ALT  2732  csbied  3874  ssrel  5733  kmlem1  10067  bnj1476  35008  bnj1533  35013  bj-alrimd  36909  bj-exlimd  36921  bj-ax12ig  36934  bj-alextruim  36950  axc11n11  36998  bj-modalbe  37004  bj-modal4  37032  bj-wnfanf  37037  bj-wnfenf  37038  bj-19.12  37039  bj-pm11.53vw  37083  mpobi123f  38500  mptbi12f  38504  ismnushort  44749  setrec2mpt  50187