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  6558  djune  7271  omp1eomlem  7287  fodjum  7339  fodju0  7340  ismkvnex  7348  mkvprop  7351  omniwomnimkv  7360  pr2cv1  7394  3nelsucpw1  7445  xrltnr  10007  nltmnf  10016  xnn0xadd0  10095  fnpr2ob  13416  2lgslem3  15823  2lgslem4  15825  structiedg0val  15884  3dom  16537  2omap  16544  pwle2  16549  nninfalllem1  16560  nninfall  16561  nninfsellemeq  16566  trirec0xor  16599