Theorem sylg 1843

Description: A syllogism combined with generalization. Inference associated with sylgt 1842. General form of alrimih 1844. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1844. (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 1831	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1815  ax-4 1829

This theorem is referenced by:  alrimih  1844  ax9ALT  2757  csbied  3888  ssrel  5755  kmlem1  10107  bnj1476  35142  bnj1533  35147  bj-alrimd  37068  bj-exlimd  37080  bj-ax12ig  37093  bj-alextruim  37109  axc11n11  37157  bj-modalbe  37163  bj-modal4  37191  bj-wnfanf  37196  bj-wnfenf  37197  bj-19.12  37198  bj-pm11.53vw  37242  mpobi123f  38661  mptbi12f  38665  ismnushort  44877  setrec2mpt  50318