Theorem bj-mpgs 36552

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 inference ⊢ 𝜑 ⇒ ⊢ 𝜓. Strong necessity is stronger than necessity, and equivalent to it when sp 2179 (modal T) is available. Therefore, this theorem is stronger than mpg 1792 when sp 2179 is not available. (Contributed by BJ, 1-Nov-2023.)

Hypotheses

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

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 1790	. 2 ⊢ ∀𝑥𝜑
3		bj-mpgs.maj	. 2 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)
4	1, 2, 3	mp2an 691	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  bj-nnfbii  36670  bj-nnfth  36685