Theorem neqne 2428

Description: From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
neqne	⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)

Proof of Theorem neqne

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝐴 = 𝐵)
2	1	neqned 2427	1 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1402   ≠ wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  prfidceq  7235  xaddf  10257  seq3f1olemstep  10966  dvdsabseq  12633  lgsval  16289  upgriswlkdc  16767  umgrclwwlkge2  16809