Theorem 3netr4g 2546

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)

Hypotheses

Ref	Expression
3netr4g.1	⊢ (φ → A ≠ B)
3netr4g.2	⊢ C = A
3netr4g.3	⊢ D = B

Assertion

Ref	Expression
3netr4g	⊢ (φ → C ≠ D)

Proof of Theorem 3netr4g

Step	Hyp	Ref	Expression
1		3netr4g.1	. 2 ⊢ (φ → A ≠ B)
2		3netr4g.2	. . 3 ⊢ C = A
3		3netr4g.3	. . 3 ⊢ D = B
4	2, 3	neeq12i 2529	. 2 ⊢ (C ≠ D ↔ A ≠ B)
5	1, 4	sylibr 203	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)