Theorem sylg 1817

Description: A syllogism combined with generalization. Inference associated with sylgt 1816. General form of alrimih 1818. (Contributed 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 1805	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1789  ax-4 1803

This theorem is referenced by:  alrimih  1818  ax9ALT  2719  raleqbidvvOLD  3322  csbied  3924  rzal  4501  ssrel  5773  ssrelOLD  5774  kmlem1  10142  bnj1476  34376  bnj1533  34381  bj-alrimd  35997  bj-exlimd  36002  bj-ax12ig  36013  axc11n11  36060  bj-modalbe  36066  bj-modal4  36092  bj-wnfanf  36097  bj-wnfenf  36098  bj-19.12  36139  bj-pm11.53vw  36154  mpobi123f  37533  mptbi12f  37537  ismnushort  43609  setrec2mpt  47989