Theorem sylg 1818

Description: A syllogism combined with generalization. Inference associated with sylgt 1817. General form of alrimih 1819. (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 1806	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1790  ax-4 1804

This theorem is referenced by:  alrimih  1819  ax9ALT  2723  raleqbidvvOLD  3327  csbied  3930  rzal  4509  ssrel  5784  ssrelOLD  5785  kmlem1  10173  bnj1476  34478  bnj1533  34483  bj-alrimd  36096  bj-exlimd  36101  bj-ax12ig  36112  axc11n11  36159  bj-modalbe  36165  bj-modal4  36191  bj-wnfanf  36196  bj-wnfenf  36197  bj-19.12  36238  bj-pm11.53vw  36253  mpobi123f  37635  mptbi12f  37639  ismnushort  43738  setrec2mpt  48128