Theorem sylg 1850

Description: A syllogism combined with generalization. Inference associated with sylgt 1849. General form of alrimih 1851. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1851. (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 1838	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 18	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1822  ax-4 1836

This theorem is referenced by:  alrimih  1851  ax9ALT  2764  csbied  3897  ssrel  5770  kmlem1  10134  bnj1476  35180  bnj1533  35185  bj-alrimd  37141  bj-exlimd  37153  bj-ax12ig  37166  bj-alextruim  37182  axc11n11  37230  bj-modalbe  37236  bj-modal4  37264  bj-wnfanf  37269  bj-wnfenf  37270  bj-19.12  37271  bj-pm11.53vw  37315  mpobi123f  38735  mptbi12f  38739  ismnushort  44937  setrec2mpt  50394