Theorem necon1addc 2443

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

Hypothesis

Ref	Expression
necon1addc.1	|- ( ph -> (DECID ps -> ( -. ps -> A = B ) ) )

Assertion

Ref	Expression
necon1addc	|- ( ph -> (DECID ps -> ( A =/= B -> ps ) ) )

Proof of Theorem necon1addc

Step	Hyp	Ref	Expression
1		df-ne 2368	. 2 |- ( A =/= B <-> -. A = B )
2		necon1addc.1	. . 3 |- ( ph -> (DECID ps -> ( -. ps -> A = B ) ) )
3		con1dc 857	. . 3 |- (DECID ps -> ( ( -. ps -> A = B ) -> ( -. A = B -> ps ) ) )
4	2, 3	sylcom 28	. 2 |- ( ph -> (DECID ps -> ( -. A = B -> ps ) ) )
5	1, 4	syl7bi 165	1 |- ( ph -> (DECID ps -> ( A =/= B -> ps ) ) )

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:  seqf1oglem1  10613  seqf1oglem2  10614