Theorem plvcofph 41113

Description: Given, a,b,d, and "definitions" for c, e, f: f is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)

Hypotheses

Ref	Expression
plvcofph.1	|- ( ch <-> ( ( ( ( ph /\ ps ) <-> ph ) -> ( ph /\ -. ( ph /\ -. ph ) ) ) /\ ( ph /\ -. ( ph /\ -. ph ) ) ) )
plvcofph.2	|- ( ta <-> ( ( ch -> th ) /\ ( ph <-> ch ) /\ ( ( ph -> ps ) -> ( ps <-> th ) ) ) )
plvcofph.3	|- ( et <-> ( ch /\ ta ) )
plvcofph.4	|- ph
plvcofph.5	|- ps
plvcofph.6	|- th

Assertion

Ref	Expression
plvcofph	|- et

Proof of Theorem plvcofph

Step	Hyp	Ref	Expression
1		plvcofph.1	. . . 4 |- ( ch <-> ( ( ( ( ph /\ ps ) <-> ph ) -> ( ph /\ -. ( ph /\ -. ph ) ) ) /\ ( ph /\ -. ( ph /\ -. ph ) ) ) )
2		plvcofph.4	. . . 4 |- ph
3		plvcofph.5	. . . 4 |- ps
4	1, 2, 3	plcofph 41111	. . 3 |- ch
5		plvcofph.2	. . . 4 |- ( ta <-> ( ( ch -> th ) /\ ( ph <-> ch ) /\ ( ( ph -> ps ) -> ( ps <-> th ) ) ) )
6		plvcofph.6	. . . 4 |- th
7	5, 2, 3, 4, 6	pldofph 41112	. . 3 |- ta
8	4, 7	pm3.2i 471	. 2 |- ( ch /\ ta )
9		plvcofph.3	. . . 4 |- ( et <-> ( ch /\ ta ) )
10	9	bicomi 214	. . 3 |- ( ( ch /\ ta ) <-> et )
11	10	biimpi 206	. 2 |- ( ( ch /\ ta ) -> et )
12	8, 11	ax-mp 5	1 |- et

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037

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-an 386  df-3an 1039

This theorem is referenced by: (None)