Theorem naim1 32384

Description: Constructor theorem for -/\. (Contributed by Anthony Hart, 1-Sep-2011.)

Assertion

Ref	Expression
naim1	|- ( ( ph -> ps ) -> ( ( ps -/\ ch ) -> ( ph -/\ ch ) ) )

Proof of Theorem naim1

Step	Hyp	Ref	Expression
1		con3 149	. . 3 |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )
2	1	orim1d 884	. 2 |- ( ( ph -> ps ) -> ( ( -. ps \/ -. ch ) -> ( -. ph \/ -. ch ) ) )
3		pm3.13 522	. . . 4 |- ( -. ( ps /\ ch ) -> ( -. ps \/ -. ch ) )
4		pm3.14 523	. . . 4 |- ( ( -. ph \/ -. ch ) -> -. ( ph /\ ch ) )
5	3, 4	imim12i 62	. . 3 |- ( ( ( -. ps \/ -. ch ) -> ( -. ph \/ -. ch ) ) -> ( -. ( ps /\ ch ) -> -. ( ph /\ ch ) ) )
6		df-nan 1448	. . 3 |- ( ( ps -/\ ch ) <-> -. ( ps /\ ch ) )
7		df-nan 1448	. . 3 |- ( ( ph -/\ ch ) <-> -. ( ph /\ ch ) )
8	5, 6, 7	3imtr4g 285	. 2 |- ( ( ( -. ps \/ -. ch ) -> ( -. ph \/ -. ch ) ) -> ( ( ps -/\ ch ) -> ( ph -/\ ch ) ) )
9	2, 8	syl 17	1 |- ( ( ph -> ps ) -> ( ( ps -/\ ch ) -> ( ph -/\ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   -/\ wnan 1447

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  df-nan 1448

This theorem is referenced by:  naim1i  32386