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Theorem sylg 1823
Description: A syllogism combined with generalization. Inference associated with sylgt 1822. General form of alrimih 1824. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1824. (Revised by BJ, 4-Oct-2019.)
Hypotheses
Ref Expression
sylg.1 (𝜑 → ∀𝑥𝜓)
sylg.2 (𝜓𝜒)
Assertion
Ref Expression
sylg (𝜑 → ∀𝑥𝜒)

Proof of Theorem sylg
StepHypRef Expression
1 sylg.1 . 2 (𝜑 → ∀𝑥𝜓)
2 sylg.2 . . 3 (𝜓𝜒)
32alimi 1811 . 2 (∀𝑥𝜓 → ∀𝑥𝜒)
41, 3syl 17 1 (𝜑 → ∀𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1795  ax-4 1809
This theorem is referenced by:  alrimih  1824  ax9ALT  2732  raleqbidvvOLD  3335  csbied  3935  rzal  4509  ssrel  5792  ssrelOLD  5793  kmlem1  10191  bnj1476  34861  bnj1533  34866  bj-alrimd  36621  bj-exlimd  36626  bj-ax12ig  36637  axc11n11  36683  bj-modalbe  36689  bj-modal4  36715  bj-wnfanf  36720  bj-wnfenf  36721  bj-19.12  36762  bj-pm11.53vw  36777  mpobi123f  38169  mptbi12f  38173  ismnushort  44320  setrec2mpt  49216
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