Theorem nesymi 2302

Description: Inference associated with nesym 2301. (Contributed by BJ, 7-Jul-2018.)

Hypothesis

Ref	Expression
nesymi.1	⊢ 𝐴 ≠ 𝐵

Assertion

Ref	Expression
nesymi	⊢ ¬ 𝐵 = 𝐴

Proof of Theorem nesymi

Step	Hyp	Ref	Expression
1		nesymi.1	. 2 ⊢ 𝐴 ≠ 𝐵
2		nesym 2301	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)
3	1, 2	mpbi 144	1 ⊢ ¬ 𝐵 = 𝐴

Syntax hints:  ¬ wn 3   = wceq 1290   ≠ wne 2256

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-5 1382  ax-gen 1384  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082  df-ne 2257

This theorem is referenced by:  frec0g  6176  djune  6823  fodjuomnilemm  6862  fodjuomnilem0  6863  xrltnr  9311  nltmnf  9319  nninfalllem1  12171  nninfall  12172  nninfsellemeq  12178