Theorem bj-mpgs 35150

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 2176 (modal T) is available. Therefore, this theorem is stronger than mpg 1799 when sp 2176 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 1797	. 2 ⊢ ∀𝑥𝜑
3		bj-mpgs.maj	. 2 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)
4	1, 2, 3	mp2an 690	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797

This theorem depends on definitions:  df-bi 206  df-an 397

This theorem is referenced by:  bj-nnfbii  35268  bj-nnfth  35283