Theorem neeq1i 2418

Description: Inference for inequality. (Contributed by NM, 29-Apr-2005.)

Hypothesis

Ref	Expression
neeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
neeq1i	⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)

Proof of Theorem neeq1i

Step	Hyp	Ref	Expression
1		neeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		neeq1 2416	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)

Syntax hints:   ↔ wb 105   = wceq 1398   ≠ wne 2403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-ne 2404

This theorem is referenced by:  neeq12i  2420  eqnetri  2426  eqnetrrid  2434  rabn0r  3523  seqf1oglem1  10825