Theorem necon3ai 2254

Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon3ai.1	|- ( ph -> A = B )

Assertion

Ref	Expression
necon3ai	|- ( A =/= B -> -. ph )

Proof of Theorem necon3ai

Step	Hyp	Ref	Expression
1		df-ne 2206	. 2 |- ( A =/= B <-> -. A = B )
2		necon3ai.1	. . 3 |- ( ph -> A = B )
3	2	con3i 562	. 2 |- ( -. A = B -> -. ph )
4	1, 3	sylbi 114	1 |- ( A =/= B -> -. ph )

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

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-in1 544  ax-in2 545

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

This theorem is referenced by:  disjsn2  3433  0nelxp  4372  fvunsng  5357