Theorem bj-mpgs 37231

Description: From a closed form theorem (the major premise) with an antecedent in the "strong necessity" modality (in the language of modal logic), deduce the associated inference. Strong necessity is stronger than necessity, and equivalent to it when sp 2218 (modal T) is available. Therefore, this theorem is stronger than mpg 1826, and strictly stronger when sp 2218 is not available. (Contributed by BJ, 1-Nov-2023.)

Hypotheses

Ref	Expression
bj-mpgs.maj	⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)
bj-mpgs.min	⊢ 𝜑

Assertion

Ref	Expression
bj-mpgs	⊢ 𝜓

Proof of Theorem bj-mpgs

Step	Hyp	Ref	Expression
1		bj-mpgs.min	. 2 ⊢ 𝜑
2	1	ax-gen 1824	. 2 ⊢ ∀𝑥𝜑
3		bj-mpgs.maj	. 2 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)
4	1, 2, 3	mp2an 704	1 ⊢ 𝜓

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824

This proof depends on definitions:  df-bi 210  df-an 401

This theorem is used by:  bj-nnfth  37397  bj-nnfbii  37402