Theorem neeq2i 2393

Description: Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Hypothesis

Ref	Expression
neeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
neeq2i	⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)

Proof of Theorem neeq2i

Step	Hyp	Ref	Expression
1		neeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		neeq2 2391	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1373   ≠ wne 2377

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 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-cleq 2199  df-ne 2378

This theorem is referenced by:  neeq12i  2394  neeqtri  2404  exmidsbthrlem  16102