Theorem neeqtrd 2364

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Hypotheses

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

Assertion

Ref	Expression
neeqtrd	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem neeqtrd

Step	Hyp	Ref	Expression
1		neeqtrd.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		neeqtrd.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
3	2	neeq2d 2355	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶))
4	1, 3	mpbid 146	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1343   ≠ wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-ne 2337

This theorem is referenced by:  neeqtrrd  2366  xaddass2  9806  modsumfzodifsn  10331