Theorem eqnetrd 2401

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

Hypotheses

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

Assertion

Ref	Expression
eqnetrd	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem eqnetrd

Step	Hyp	Ref	Expression
1		eqnetrd.2	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
2		eqnetrd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	neeq1d 2395	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
4	1, 3	mpbird 167	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = 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:  eqnetrrd  2403  ifnetruedc  3618  ifnefals  3619  frecabcl  6498  frecsuclem  6505  omp1eomlem  7211  xaddnemnf  9999  xaddnepnf  10000  hashprg  10975  bezoutr1  12429  phibndlem  12613  dfphi2  12617  lgsne0  15590  2sqlem8a  15674  2sqlem8  15675