Theorem necon1bddc 2444

Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)

Hypothesis

Ref	Expression
necon1bddc.1	|- ( ph -> (DECID A = B -> ( A =/= B -> ps ) ) )

Assertion

Ref	Expression
necon1bddc	|- ( ph -> (DECID A = B -> ( -. ps -> A = B ) ) )

Proof of Theorem necon1bddc

Step	Hyp	Ref	Expression
1		necon1bddc.1	. . 3 |- ( ph -> (DECID A = B -> ( A =/= B -> ps ) ) )
2		df-ne 2368	. . . 4 |- ( A =/= B <-> -. A = B )
3	2	imbi1i 238	. . 3 |- ( ( A =/= B -> ps ) <-> ( -. A = B -> ps ) )
4	1, 3	imbitrdi 161	. 2 |- ( ph -> (DECID A = B -> ( -. A = B -> ps ) ) )
5		con1dc 857	. 2 |- (DECID A = B -> ( ( -. A = B -> ps ) -> ( -. ps -> A = B ) ) )
6	4, 5	sylcom 28	1 |- ( ph -> (DECID A = B -> ( -. ps -> A = B ) ) )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 835   = wceq 1364   =/= wne 2367

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-ne 2368

This theorem is referenced by:  necon1ddc  2445  pc2dvds  12509  perfectlem2  15246