Theorem sylg 1824

Description: A syllogism combined with generalization. Inference associated with sylgt 1823. General form of alrimih 1825. (Contributed 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 1813	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1797  ax-4 1811

This theorem is referenced by:  alrimih  1825  ax9ALT  2753  rzal  4401  ssrel  5626  kmlem1  9610  bnj1476  32347  bnj1533  32352  bj-alrimd  34347  bj-exlimd  34352  bj-ax12ig  34363  axc11n11  34410  bj-modalbe  34416  bj-modal4  34442  bj-wnfanf  34447  bj-wnfenf  34448  bj-19.12  34486  mpobi123f  35880  mptbi12f  35884