Theorem neqcomd 2747

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 2744	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
3	1, 2	sylnib 327	1 ⊢ (𝜑 → ¬ 𝐵 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-cleq 2729

This theorem is referenced by:  phpeqd  9055  phpeqdOLD  9065  simpgnsgd  19770  aks4d1p8d2  40305  rr-phpd  42049