Theorem sylg 1824

Description: A syllogism combined with generalization. Inference associated with sylgt 1823. General form of alrimih 1825. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1825. (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 1812	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1796  ax-4 1810

This theorem is referenced by:  alrimih  1825  ax9ALT  2731  raleqbidvvOLD  3305  csbied  3885  ssrel  5732  kmlem1  10061  bnj1476  35003  bnj1533  35008  bj-alrimd  36820  bj-exlimd  36825  bj-ax12ig  36836  axc11n11  36883  bj-modalbe  36889  bj-modal4  36915  bj-wnfanf  36920  bj-wnfenf  36921  bj-19.12  36962  bj-pm11.53vw  36977  mpobi123f  38363  mptbi12f  38367  ismnushort  44542  setrec2mpt  49942