Theorem 3netr4g 3023

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 3010	. 2 ⊢ (𝐶 ≠ 𝐷 ↔ 𝐴 ≠ 𝐵)
5	1, 4	sylibr 233	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:   → wi 4   = wceq 1539   ≠ wne 2943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-ne 2944

This theorem is referenced by:  aalioulem2  25493  mapdpglem18  39703  line2x  46100