Theorem sylg 1853

Description: A syllogism combined with generalization. Inference associated with sylgt 1852. General form of alrimih 1854. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1854. (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 1841	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 18	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1825  ax-4 1839

This theorem is referenced by:  alrimih  1854  ax9ALT  2758  csbied  3889  ssrel  5769  kmlem1  10130  bnj1476  35235  bnj1533  35240  bj-alrimd  37238  bj-exlimd  37250  bj-ax12ig  37263  bj-alextruim  37279  axc11n11  37327  bj-modalbe  37333  bj-modal4  37361  bj-wnfanf  37366  bj-wnfenf  37367  bj-19.12  37368  bj-pm11.53vw  37412  mpobi123f  38831  mptbi12f  38835  ismnushort  45031  setrec2mpt  50495