Theorem prvlem2 32587

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

Hypothesis

Ref	Expression
prvlem2.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
prvlem2	|- (Prv ph -> (Prv ps -> Prv ch ) )

Proof of Theorem prvlem2

Step	Hyp	Ref	Expression
1		prvlem2.1	. . 3 |- ( ph -> ( ps -> ch ) )
2	1	prvlem1 32586	. 2 |- (Prv ph -> Prv ( ps -> ch ) )
3		ax-prv2 32584	. 2 |- (Prv ( ps -> ch ) -> (Prv ps -> Prv ch ) )
4	2, 3	syl 17	1 |- (Prv ph -> (Prv ps -> Prv ch ) )

Syntax hints:   -> wi 4  Prv cprvb 32582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-prv1 32583  ax-prv2 32584

This theorem is referenced by:  bj-babygodel  32588  bj-babylob  32589