Theorem 3netr3d 2277

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

Hypotheses

Ref	Expression
3netr3d.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
3netr3d.2	⊢ (𝜑 → 𝐴 = 𝐶)
3netr3d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3netr3d	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Proof of Theorem 3netr3d

Step	Hyp	Ref	Expression
1		3netr3d.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		3netr3d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
3		3netr3d.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 3	neeq12d 2265	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
5	1, 4	mpbid 145	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:   → wi 4   = wceq 1284   ≠ wne 2245

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 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-ne 2246

This theorem is referenced by: (None)