Theorem prvlem1 37222

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 37219	. 2 ⊢ Prv (𝜑 → 𝜓)
3		ax-prv2 37220	. 2 ⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
4	2, 3	ax-mp 5	1 ⊢ (Prv 𝜑 → Prv 𝜓)

This proof depends on syntax axioms:   → wi 4  Prv cprvb 37218

This proof depends on axioms:  ax-mp 5  ax-prv1 37219  ax-prv2 37220

This theorem is used by:  prvlem2  37223  bj-babygodel  37224