Theorem neeq2d 2396

Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Hypothesis

Ref	Expression
neeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
neeq2d	⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))

Proof of Theorem neeq2d

Step	Hyp	Ref	Expression
1		neeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		neeq2 2391	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373   ≠ wne 2377

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 615  ax-in2 616  ax-5 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-cleq 2199  df-ne 2378

This theorem is referenced by:  neeq12d  2397  neeqtrd  2405  sqrt2irr  12559  ennnfonelemk  12846  ennnfoneleminc  12857  ennnfonelemex  12860  ennnfonelemnn0  12868  ennnfonelemr  12869  setscomd  12948