Theorem necomi 2421

Description: Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)

Hypothesis

Ref	Expression
necomi.1	⊢ 𝐴 ≠ 𝐵

Assertion

Ref	Expression
necomi	⊢ 𝐵 ≠ 𝐴

Proof of Theorem necomi

Step	Hyp	Ref	Expression
1		necomi.1	. 2 ⊢ 𝐴 ≠ 𝐵
2		necom 2420	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
3	1, 2	mpbi 144	1 ⊢ 𝐵 ≠ 𝐴

Syntax hints:   ≠ 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-5 1435  ax-gen 1437  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-ne 2337

This theorem is referenced by:  0nep0  4144  xp01disj  6401  xp01disjl  6402  djulclb  7020  djuinr  7028  pnfnemnf  7953  mnfnepnf  7954  ltneii  7995  1ne0  8925  0ne2  9062  fzprval  10017  0tonninf  10374  1tonninf  10375