neeq1i - New Foundations Explorer

	New Foundations Explorer		< Previous Next > Nearby theorems
Mirrors > Home > NFE Home > Th. List > neeq1i		GIF version

Theorem neeq1i 2526

Description: Inference for inequality. (Contributed by NM, 29-Apr-2005.)

**Hypothesis**
Ref	Expression
neeq1i.1	⊢ A = B

**Assertion**
Ref	Expression
neeq1i	⊢ (A ≠ C ↔ B ≠ C)

**Proof of Theorem neeq1i**
Step	Hyp	Ref	Expression
1		neeq1i.1	. 2 ⊢ A = B
2		neeq1 2524	. 2 ⊢ (A = B → (A ≠ C ↔ B ≠ C))
3	1, 2	ax-mp 8	1 ⊢ (A ≠ C ↔ B ≠ C)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 176 = wceq 1642 ≠ wne 2516

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-ex 1542 df-cleq 2346 df-ne 2518

This theorem is referenced by: neeq12i 2528 eqnetri 2533 syl5eqner 2541 rabn0 3570 ltfinirr 4457 evenodddisj 4516