Theorem neeq2i 2356

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 2354	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)

Syntax hints:   ↔ wb 104   = wceq 1348   ≠ wne 2340

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-ne 2341

This theorem is referenced by:  neeq12i  2357  neeqtri  2367  exmidsbthrlem  14054