Theorem necon4addc 2434

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 2365	. . . 4 |- ( A =/= B <-> -. A = B )
3	2	imbi1i 238	. . 3 |- ( ( A =/= B -> -. ps ) <-> ( -. A = B -> -. ps ) )
4		condc 854	. . 3 |- (DECID A = B -> ( ( -. A = B -> -. ps ) -> ( ps -> A = B ) ) )
5	3, 4	biimtrid 152	. 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 835   = wceq 1364   =/= wne 2364

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 2365

This theorem is referenced by:  4sqlem11  12539