Theorem neeq2i 3025

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 2778	. 2 ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)
3	2	necon3bii 3012	1 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ≠ wne 2960

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 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-ne 2961

This theorem is used by:  neeqtri  3032  omsucne  7887  suppvalbr  8166  nosgnn0  27873  upgr3v3e3cycl  30602  upgr4cycl4dv4e  30607  disjdsct  33119  divnumden2  33230  usgrgt2cycl  35667  onov0suclim  44059