Theorem neeq2i 2997

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 2748	. 2 ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)
3	2	necon3bii 2984	1 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1540   ≠ wne 2932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-ne 2933

This theorem is referenced by:  neeqtri  3004  omsucne  7878  suppvalbr  8161  nosgnn0  27620  upgr3v3e3cycl  30107  upgr4cycl4dv4e  30112  disjdsct  32626  divnumden2  32740  usgrgt2cycl  35098  onov0suclim  43245