Theorem 3netr3g 3038

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

Hypotheses

Ref	Expression
3netr3g.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
3netr3g.2	⊢ 𝐴 = 𝐶
3netr3g.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3netr3g	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Proof of Theorem 3netr3g

Step	Hyp	Ref	Expression
1		3netr3g.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		3netr3g.2	. . 3 ⊢ 𝐴 = 𝐶
3		3netr3g.3	. . 3 ⊢ 𝐵 = 𝐷
4	2, 3	neeq12i 3026	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)
5	1, 4	sylib 221	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2960

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-ne 2961

This theorem is used by: (None)