Theorem nesymir 2414

Description: Inference associated with nesym 2412. (Contributed by BJ, 7-Jul-2018.)

Hypothesis

Ref	Expression
nesymir.1	⊢ ¬ 𝐴 = 𝐵

Assertion

Ref	Expression
nesymir	⊢ 𝐵 ≠ 𝐴

Proof of Theorem nesymir

Step	Hyp	Ref	Expression
1		nesymir.1	. 2 ⊢ ¬ 𝐴 = 𝐵
2		nesym 2412	. 2 ⊢ (𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbir 146	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: (None)