Theorem nesymi 2410

Description: Inference associated with nesym 2409. (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 2409	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ¬ 𝐵 = 𝐴

Syntax hints:  ¬ wn 3   = wceq 1364   ≠ wne 2364

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 1458  ax-gen 1460  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-ne 2365

This theorem is referenced by:  frec0g  6450  djune  7137  omp1eomlem  7153  fodjum  7205  fodju0  7206  ismkvnex  7214  mkvprop  7217  omniwomnimkv  7226  3nelsucpw1  7294  xrltnr  9845  nltmnf  9854  xnn0xadd0  9933  fnpr2ob  12923  pwle2  15489  nninfalllem1  15498  nninfall  15499  nninfsellemeq  15504  trirec0xor  15535