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  2756  csbied  3883  ssrel  5759  kmlem1  10229  bnj1476  35477  bnj1533  35482  bj-alrimd  37495  bj-exlimd  37507  bj-ax12ig  37520  bj-alextruim  37536  axc11n11  37584  bj-modalbe  37590  bj-modal4  37618  bj-wnfanf  37623  bj-wnfenf  37624  bj-19.12  37625  bj-pm11.53vw  37669  mpobi123f  39094  mptbi12f  39098  ismnushort  45284  setrec2mpt  50789