Theorem sylg 1856

Description: A syllogism combined with generalization. Inference associated with sylgt 1855. General form of alrimih 1857. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1857. (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 1844	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 18	1 ⊢ (𝜑 → ∀𝑥𝜒)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1828  ax-4 1842

This theorem is used by:  alrimih  1857  ax9ALT  2755  csbied  3883  ssrel  5763  kmlem1  10156  bnj1476  35359  bnj1533  35364  bj-alrimd  37329  bj-exlimd  37341  bj-ax12ig  37354  bj-alextruim  37370  axc11n11  37418  bj-modalbe  37424  bj-modal4  37452  bj-wnfanf  37457  bj-wnfenf  37458  bj-19.12  37459  bj-pm11.53vw  37503  mpobi123f  38913  mptbi12f  38917  ismnushort  45128  setrec2mpt  50626