Theorem prvlem1 32586

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

Hypothesis

Ref	Expression
prvlem1.1	|- ( ph -> ps )

Assertion

Ref	Expression
prvlem1	|- (Prv ph -> Prv ps )

Proof of Theorem prvlem1

Step	Hyp	Ref	Expression
1		prvlem1.1	. . 3 |- ( ph -> ps )
2	1	ax-prv1 32583	. 2 |- Prv ( ph -> ps )
3		ax-prv2 32584	. 2 |- (Prv ( ph -> ps ) -> (Prv ph -> Prv ps ) )
4	2, 3	ax-mp 5	1 |- (Prv ph -> Prv ps )

Syntax hints:   -> wi 4  Prv cprvb 32582

This theorem was proved from axioms:  ax-mp 5  ax-prv1 32583  ax-prv2 32584

This theorem is referenced by:  prvlem2  32587  bj-babygodel  32588