Theorem necon4bddc 2276

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

Hypothesis

Ref	Expression
necon4bddc.1	|- ( ph -> (DECID ps -> ( -. ps -> A =/= B ) ) )

Assertion

Ref	Expression
necon4bddc	|- ( ph -> (DECID ps -> ( A = B -> ps ) ) )

Proof of Theorem necon4bddc

Step	Hyp	Ref	Expression
1		necon4bddc.1	. . 3 |- ( ph -> (DECID ps -> ( -. ps -> A =/= B ) ) )
2		df-ne 2206	. . 3 |- ( A =/= B <-> -. A = B )
3	1, 2	syl8ib 155	. 2 |- ( ph -> (DECID ps -> ( -. ps -> -. A = B ) ) )
4		condc 749	. 2 |- (DECID ps -> ( ( -. ps -> -. A = B ) -> ( A = B -> ps ) ) )
5	3, 4	sylcom 25	1 |- ( ph -> (DECID ps -> ( A = B -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 742   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206

This theorem is referenced by: (None)