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Theorem prvlem1 32255
Description: An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
Hypothesis
Ref Expression
prvlem1.1 (𝜑𝜓)
Assertion
Ref Expression
prvlem1 (Prv 𝜑 → Prv 𝜓)

Proof of Theorem prvlem1
StepHypRef Expression
1 prvlem1.1 . . 3 (𝜑𝜓)
21ax-prv1 32252 . 2 Prv (𝜑𝜓)
3 ax-prv2 32253 . 2 (Prv (𝜑𝜓) → (Prv 𝜑 → Prv 𝜓))
42, 3ax-mp 5 1 (Prv 𝜑 → Prv 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  Prv cprvb 32251
This theorem was proved from axioms:  ax-mp 5  ax-prv1 32252  ax-prv2 32253
This theorem is referenced by:  prvlem2  32256  bj-babygodel  32257
  Copyright terms: Public domain W3C validator