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Theorem minimp-ax2 1564
Description: Derivation of ax-2 7 from ax-mp 5 and minimp 1560. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
minimp-ax2 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
Proof of Theorem minimp-ax2
StepHypRef Expression
1 minimp-ax2c 1563 . 2 |- ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) )
2 minimp-ax2c 1563 . . 3 |- ( ( ( ph -> ps ) -> ( ph -> ( ps -> ch ) ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
3 minimp-sylsimp 1561 . . 3 |- ( ( ( ( ph -> ps ) -> ( ph -> ( ps -> ch ) ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) )
42, 3ax-mp 5 . 2 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
5 minimp-ax2c 1563 . . 3 |- ( ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) ) -> ( ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) )
6 minimp-sylsimp 1561 . . 3 |- ( ( ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) ) -> ( ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) ) )
75, 6ax-mp 5 . 2 |- ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ( ph -> ( ps -> ch ) ) -> ( ( ( ph -> ps ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) )
81, 4, 7mp2 9 1 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7
This theorem is referenced by:  minimp-pm2.43  1565
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