Theorem neqcomd 2144

Description: Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypothesis

Ref	Expression
neqcomd.1	⊢ (𝜑 → ¬ 𝐴 = 𝐵)

Assertion

Ref	Expression
neqcomd	⊢ (𝜑 → ¬ 𝐵 = 𝐴)

Proof of Theorem neqcomd

Step	Hyp	Ref	Expression
1		neqcomd.1	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
2		eqcom 2141	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
3	1, 2	sylnib 665	1 ⊢ (𝜑 → ¬ 𝐵 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132

This theorem is referenced by: (None)