Theorem necon4addc 2406

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)

Hypothesis

Ref	Expression
necon4addc.1	|- ( ph -> (DECID A = B -> ( A =/= B -> -. ps ) ) )

Assertion

Ref	Expression
necon4addc	|- ( ph -> (DECID A = B -> ( ps -> A = B ) ) )

Proof of Theorem necon4addc

Step	Hyp	Ref	Expression
1		necon4addc.1	. 2 |- ( ph -> (DECID A = B -> ( A =/= B -> -. ps ) ) )
2		df-ne 2337	. . . 4 |- ( A =/= B <-> -. A = B )
3	2	imbi1i 237	. . 3 |- ( ( A =/= B -> -. ps ) <-> ( -. A = B -> -. ps ) )
4		condc 843	. . 3 |- (DECID A = B -> ( ( -. A = B -> -. ps ) -> ( ps -> A = B ) ) )
5	3, 4	syl5bi 151	. 2 |- (DECID A = B -> ( ( A =/= B -> -. ps ) -> ( ps -> A = B ) ) )
6	1, 5	sylcom 28	1 |- ( ph -> (DECID A = B -> ( ps -> A = B ) ) )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 824   = wceq 1343   =/= wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-ne 2337

This theorem is referenced by: (None)