Theorem bj-peirce 32543

Description: Proof of peirce 193 from minimal implicational calculus, the axiomatic definition of disjunction (olc 399, orc 400, jao 534), and Curry's axiom bj-curry 32542. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-peirce	|- ( ( ( ph -> ps ) -> ph ) -> ph )

Proof of Theorem bj-peirce

Step	Hyp	Ref	Expression
1		bj-curry 32542	. . 3 |- ( ph \/ ( ph -> ps ) )
2		bj-orim2 32541	. . 3 |- ( ( ( ph -> ps ) -> ph ) -> ( ( ph \/ ( ph -> ps ) ) -> ( ph \/ ph ) ) )
3	1, 2	mpi 20	. 2 |- ( ( ( ph -> ps ) -> ph ) -> ( ph \/ ph ) )
4		pm1.2 535	. 2 |- ( ( ph \/ ph ) -> ph )
5	3, 4	syl 17	1 |- ( ( ( ph -> ps ) -> ph ) -> ph )

Syntax hints:   -> wi 4   \/ wo 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386

This theorem is referenced by: (None)