Theorem 3netr3d 2543

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 2532	. 2 ⊢ (φ → (A ≠ B ↔ C ≠ D))
5	1, 4	mpbid 201	1 ⊢ (φ → C ≠ D)

This proof depends on syntax axioms:   → wi 4   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is used by: (None)