Theorem 3netr4d 2282

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Hypotheses

Ref	Expression
3netr4d.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
3netr4d.2	⊢ (𝜑 → 𝐶 = 𝐴)
3netr4d.3	⊢ (𝜑 → 𝐷 = 𝐵)

Assertion

Ref	Expression
3netr4d	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Proof of Theorem 3netr4d

Step	Hyp	Ref	Expression
1		3netr4d.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		3netr4d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐴)
3		3netr4d.3	. . 3 ⊢ (𝜑 → 𝐷 = 𝐵)
4	2, 3	neeq12d 2269	. 2 ⊢ (𝜑 → (𝐶 ≠ 𝐷 ↔ 𝐴 ≠ 𝐵))
5	1, 4	mpbird 165	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:   → wi 4   = wceq 1285   ≠ wne 2249

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-ne 2250

This theorem is referenced by:  modsumfzodifsn  9530