Theorem neir 2423

Description: Inference associated with df-ne 2421. (Contributed by BJ, 7-Jul-2018.)

Hypothesis

Ref	Expression
neir.1	|- -. A = B

Assertion

Ref	Expression
neir	|- A =/= B

Proof of Theorem neir

Step	Hyp	Ref	Expression
1		neir.1	. 2 |- -. A = B
2		df-ne 2421	. 2 |- ( A =/= B <-> -. A = B )
3	1, 2	mpbir 146	1 |- A =/= B

This proof depends on syntax axioms:   -. wn 3   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  exmidonfinlem  7545  pw1ne1  7588  pw1ne3  7589  sucpw1nel3  7592  ine0  8721  pwle2  17028