Theorem prvlem2 34686

Description: An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)

Hypothesis

Ref	Expression
prvlem2.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
prvlem2	⊢ (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))

Proof of Theorem prvlem2

Step	Hyp	Ref	Expression
1		prvlem2.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	prvlem1 34685	. 2 ⊢ (Prv 𝜑 → Prv (𝜓 → 𝜒))
3		ax-prv2 34683	. 2 ⊢ (Prv (𝜓 → 𝜒) → (Prv 𝜓 → Prv 𝜒))
4	2, 3	syl 17	1 ⊢ (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))

Syntax hints:   → wi 4  Prv cprvb 34681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-prv1 34682  ax-prv2 34683

This theorem is referenced by:  bj-babygodel  34687  bj-babylob  34688