Theorem eqnetrrid 2451

Description: B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)

Hypotheses

Ref	Expression
eqnetrrid.1	⊢ 𝐵 = 𝐴
eqnetrrid.2	⊢ (𝜑 → 𝐵 ≠ 𝐶)

Assertion

Ref	Expression
eqnetrrid	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem eqnetrrid

Step	Hyp	Ref	Expression
1		eqnetrrid.2	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
2		eqnetrrid.1	. . 3 ⊢ 𝐵 = 𝐴
3	2	neeq1i 2435	. 2 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶)
4	1, 3	sylib 122	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1402   ≠ wne 2420

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is referenced by: (None)