Theorem merco2 14203

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

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 14180.

Assertion

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

Proof of Theorem merco2

Step	Hyp	Ref	Expression
1		FiA 14104	. . . . . 6 |- ( F. -> ch)
2		pm2.04 34	. . . . . 6 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> (( F. -> ch) -> ((ph -> ps) -> th)))
3	1, 2	mpi 55	. . . . 5 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> ((ph -> ps) -> th))
4		pm2.21 92	. . . . . . 7 |- (-. ph -> (ph -> ps))
5	4	imim1i 19	. . . . . 6 |- (((ph -> ps) -> th) -> (-. ph -> th))
6		idd 75	. . . . . 6 |- (((ph -> ps) -> th) -> (th -> th))
7	5, 6	jad 156	. . . . 5 |- (((ph -> ps) -> th) -> ((ph -> th) -> th))
8		looinv 99	. . . . 5 |- (((ph -> th) -> th) -> ((th -> ph) -> ph))
9	3, 7, 8	3syl 24	. . . 4 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> ((th -> ph) -> ph))
10	9	a1dd 53	. . 3 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> ((th -> ph) -> (ta -> ph)))
11	10	a1i 8	. 2 |- (et -> (((ph -> ps) -> (( F. -> ch) -> th)) -> ((th -> ph) -> (ta -> ph))))
12	11	com4l 43	1 |- (((ph -> ps) -> (( F. -> ch) -> th)) -> ((th -> ph) -> (ta -> (et -> ph))))

Syntax hints:  -. wn 2   -> wi 3   F. wfal 1261

This theorem is referenced by:  mercolem1 14204  mercolem2 14205  mercolem3 14206  mercolem4 14207  mercolem5 14208  mercolem6 14209  mercolem7 14210  mercolem8 14211  re1tbw4 14215

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-tru 1262  df-fal 1263

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