Theorem neeq2i 2998

Description: Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)

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	. . 3 ⊢ 𝐴 = 𝐵
2	1	eqeq2i 2737	. 2 ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)
3	2	necon3bii 2985	1 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)

Syntax hints:   ↔ wb 205   = wceq 1533   ≠ wne 2932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2716  df-ne 2933

This theorem is referenced by:  neeqtri  3005  omsucne  7868  suppvalbr  8145  nosgnn0  27532  upgr3v3e3cycl  29928  upgr4cycl4dv4e  29933  disjdsct  32419  divnumden2  32517  usgrgt2cycl  34639  onov0suclim  42574