Theorem sylg 1830

Description: A syllogism combined with generalization. Inference associated with sylgt 1829. General form of alrimih 1831. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1831. (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 1818	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1802  ax-4 1816

This theorem is referenced by:  alrimih  1831  ax9ALT  2734  csbied  3867  ssrel  5726  kmlem1  10064  bnj1476  35029  bnj1533  35034  bj-alrimd  36936  bj-exlimd  36948  bj-ax12ig  36961  bj-alextruim  36977  axc11n11  37025  bj-modalbe  37031  bj-modal4  37059  bj-wnfanf  37064  bj-wnfenf  37065  bj-19.12  37066  bj-pm11.53vw  37110  mpobi123f  38529  mptbi12f  38533  ismnushort  44745  setrec2mpt  50187