Theorem necon1aidc 2396

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon1aidc.1	|- (DECID ph -> ( -. ph -> A = B ) )

Assertion

Ref	Expression
necon1aidc	|- (DECID ph -> ( A =/= B -> ph ) )

Proof of Theorem necon1aidc

Step	Hyp	Ref	Expression
1		df-ne 2346	. 2 |- ( A =/= B <-> -. A = B )
2		necon1aidc.1	. . 3 |- (DECID ph -> ( -. ph -> A = B ) )
3		con1dc 856	. . 3 |- (DECID ph -> ( ( -. ph -> A = B ) -> ( -. A = B -> ph ) ) )
4	2, 3	mpd 13	. 2 |- (DECID ph -> ( -. A = B -> ph ) )
5	1, 4	biimtrid 152	1 |- (DECID ph -> ( A =/= B -> ph ) )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 834   = wceq 1353   =/= wne 2345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-ne 2346

This theorem is referenced by:  necon1idc  2398  lgsne0  14008