Theorem neeq2i 3006

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 2745	. 2 ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)
3	2	necon3bii 2993	1 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)

Syntax hints:   ↔ wb 205   = wceq 1541   ≠ wne 2940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724  df-ne 2941

This theorem is referenced by:  neeqtri  3013  omsucne  7870  suppvalbr  8146  nosgnn0  27150  upgr3v3e3cycl  29422  upgr4cycl4dv4e  29427  disjdsct  31911  divnumden2  32011  usgrgt2cycl  34109  onov0suclim  42009