Theorem prvlem1 34685

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

Hypothesis

Ref	Expression
prvlem1.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
prvlem1	⊢ (Prv 𝜑 → Prv 𝜓)

Proof of Theorem prvlem1

Step	Hyp	Ref	Expression
1		prvlem1.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	ax-prv1 34682	. 2 ⊢ Prv (𝜑 → 𝜓)
3		ax-prv2 34683	. 2 ⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
4	2, 3	ax-mp 5	1 ⊢ (Prv 𝜑 → Prv 𝜓)

Syntax hints:   → wi 4  Prv cprvb 34681

This theorem was proved from axioms:  ax-mp 5  ax-prv1 34682  ax-prv2 34683

This theorem is referenced by:  prvlem2  34686  bj-babygodel  34687