Theorem necon1bidc 2297

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

Hypothesis

Ref	Expression
necon1bidc.1	|- (DECID A = B -> ( A =/= B -> ph ) )

Assertion

Ref	Expression
necon1bidc	|- (DECID A = B -> ( -. ph -> A = B ) )

Proof of Theorem necon1bidc

Step	Hyp	Ref	Expression
1		df-ne 2246	. . 3 |- ( A =/= B <-> -. A = B )
2		necon1bidc.1	. . 3 |- (DECID A = B -> ( A =/= B -> ph ) )
3	1, 2	syl5bir 151	. 2 |- (DECID A = B -> ( -. A = B -> ph ) )
4		con1dc 786	. 2 |- (DECID A = B -> ( ( -. A = B -> ph ) -> ( -. ph -> A = B ) ) )
5	3, 4	mpd 13	1 |- (DECID A = B -> ( -. ph -> A = B ) )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 775   = wceq 1284   =/= wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776  df-ne 2246

This theorem is referenced by: (None)