Theorem merco2 1661

Description: A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1638. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco2	|- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( th -> ph ) -> ( ta -> ( et -> ph ) ) ) )

Proof of Theorem merco2

Step	Hyp	Ref	Expression
1		falim 1498	. . . . . 6 |- ( F. -> ch )
2		pm2.04 90	. . . . . 6 |- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( F. -> ch ) -> ( ( ph -> ps ) -> th ) ) )
3	1, 2	mpi 20	. . . . 5 |- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( ph -> ps ) -> th ) )
4		jarl 175	. . . . . 6 |- ( ( ( ph -> ps ) -> th ) -> ( -. ph -> th ) )
5		idd 24	. . . . . 6 |- ( ( ( ph -> ps ) -> th ) -> ( th -> th ) )
6	4, 5	jad 174	. . . . 5 |- ( ( ( ph -> ps ) -> th ) -> ( ( ph -> th ) -> th ) )
7		looinv 194	. . . . 5 |- ( ( ( ph -> th ) -> th ) -> ( ( th -> ph ) -> ph ) )
8	3, 6, 7	3syl 18	. . . 4 |- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( th -> ph ) -> ph ) )
9	8	a1dd 50	. . 3 |- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( th -> ph ) -> ( ta -> ph ) ) )
10	9	a1i 11	. 2 |- ( et -> ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( th -> ph ) -> ( ta -> ph ) ) ) )
11	10	com4l 92	1 |- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( th -> ph ) -> ( ta -> ( et -> ph ) ) ) )

Syntax hints:   -> wi 4   F. wfal 1488

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-tru 1486  df-fal 1489

This theorem is referenced by:  mercolem1  1662  mercolem2  1663  mercolem3  1664  mercolem4  1665  mercolem5  1666  mercolem6  1667  mercolem7  1668  mercolem8  1669  re1tbw4  1673