Theorem nesymi 2413

Description: Inference associated with nesym 2412. (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 2412	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ¬ 𝐵 = 𝐴

Syntax hints:  ¬ wn 3   = wceq 1364   ≠ wne 2367

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ne 2368

This theorem is referenced by:  frec0g  6455  djune  7144  omp1eomlem  7160  fodjum  7212  fodju0  7213  ismkvnex  7221  mkvprop  7224  omniwomnimkv  7233  3nelsucpw1  7301  xrltnr  9854  nltmnf  9863  xnn0xadd0  9942  fnpr2ob  12983  2lgslem3  15342  2lgslem4  15344  pwle2  15643  nninfalllem1  15652  nninfall  15653  nninfsellemeq  15658  trirec0xor  15689