Theorem necomi 2425

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 2424	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
3	1, 2	mpbi 144	1 ⊢ 𝐵 ≠ 𝐴

Syntax hints:   ≠ wne 2340

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-ne 2341

This theorem is referenced by:  0nep0  4151  xp01disj  6412  xp01disjl  6413  djulclb  7032  djuinr  7040  pnfnemnf  7974  mnfnepnf  7975  ltneii  8016  1ne0  8946  0ne2  9083  fzprval  10038  0tonninf  10395  1tonninf  10396