Theorem merco2 1501

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

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1478. (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 1328	. . . . . 6 |- ( F. -> ch)
2		pm2.04 76	. . . . . 6 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> (( F. -> ch) -> ((ph -> ps) -> th)))
3	1, 2	mpi 16	. . . . 5 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> ((ph -> ps) -> th))
4		jarl 155	. . . . . 6 |- (((ph -> ps) -> th) -> (-. ph -> th))
5		idd 21	. . . . . 6 |- (((ph -> ps) -> th) -> (th -> th))
6	4, 5	jad 154	. . . . 5 |- (((ph -> ps) -> th) -> ((ph -> th) -> th))
7		looinv 174	. . . . 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 42	. . 3 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> ((th -> ph) -> (ta -> ph)))
10	9	a1i 10	. 2 |- (et -> (((ph -> ps) -> (( F. -> ch) -> th)) -> ((th -> ph) -> (ta -> ph))))
11	10	com4l 78	1 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> ((th -> ph) -> (ta -> (et -> ph))))

Syntax hints:   -> wi 4   F. wfal 1317

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 177  df-tru 1319  df-fal 1320

This theorem is referenced by:  mercolem1  1502  mercolem2  1503  mercolem3  1504  mercolem4  1505  mercolem5  1506  mercolem6  1507  mercolem7  1508  mercolem8  1509  re1tbw4  1513