Theorem bj-mpgs 37065

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 2221 (modal T) is available. Therefore, this theorem is stronger than mpg 1820, and strictly stronger when sp 2221 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 1818	. 2 ⊢ ∀𝑥𝜑
3		bj-mpgs.maj	. 2 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)
4	1, 2, 3	mp2an 704	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818

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

This theorem is referenced by:  bj-nnfth  37231  bj-nnfbii  37236