Theorem nesymi 2446

Description: Inference associated with nesym 2445. (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 2445	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ¬ 𝐵 = 𝐴

Syntax hints:  ¬ wn 3   = wceq 1395   ≠ wne 2400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-ne 2401

This theorem is referenced by:  frec0g  6549  djune  7253  omp1eomlem  7269  fodjum  7321  fodju0  7322  ismkvnex  7330  mkvprop  7333  omniwomnimkv  7342  pr2cv1  7376  3nelsucpw1  7427  xrltnr  9983  nltmnf  9992  xnn0xadd0  10071  fnpr2ob  13381  2lgslem3  15788  2lgslem4  15790  structiedg0val  15849  3dom  16381  2omap  16388  pwle2  16393  nninfalllem1  16404  nninfall  16405  nninfsellemeq  16410  trirec0xor  16443