Theorem neqcomd 2776

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 2773	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
3	1, 2	sylnib 331	1 ⊢ (𝜑 → ¬ 𝐵 = 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2758

This theorem is used by:  ssnelpss  4072  phpeqd  9206  simpgnsgd  20203  selvvvval  22330  qdiff  38012  aks4d1p8d2  42893  sn-mullt0d  43300  rr-phpd  44974