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  7256  omp1eomlem  7272  fodjum  7324  fodju0  7325  ismkvnex  7333  mkvprop  7336  omniwomnimkv  7345  pr2cv1  7379  3nelsucpw1  7430  xrltnr  9987  nltmnf  9996  xnn0xadd0  10075  fnpr2ob  13389  2lgslem3  15796  2lgslem4  15798  structiedg0val  15857  3dom  16439  2omap  16446  pwle2  16451  nninfalllem1  16462  nninfall  16463  nninfsellemeq  16468  trirec0xor  16501