Theorem neeq2d 2431

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 2426	. 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 1398   ≠ wne 2412

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 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-ne 2413

This theorem is referenced by:  neeq12d  2432  neeqtrd  2440  sqrt2irr  12859  ennnfonelemk  13151  ennnfoneleminc  13162  ennnfonelemex  13165  ennnfonelemnn0  13173  ennnfonelemr  13174  setscomd  13253