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Theorem neeq1d 2529

Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

**Hypothesis**
Ref	Expression
neeq1d.1	⊢ (φ → A = B)

**Assertion**
Ref	Expression
neeq1d	⊢ (φ → (A ≠ C ↔ B ≠ C))

**Proof of Theorem neeq1d**
Step	Hyp	Ref	Expression
1		neeq1d.1	. 2 ⊢ (φ → A = B)
2		neeq1 2524	. 2 ⊢ (A = B → (A ≠ C ↔ B ≠ C))
3	1, 2	syl 15	1 ⊢ (φ → (A ≠ C ↔ B ≠ C))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ 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: neeq12d 2531 eqnetrd 2534 prnzg 3836 preaddccan2lem1 4454 preaddccan2 4455 evenodddisj 4516 vfinncvntnn 4548 ereldm 5971 map0 6025 ce0addcnnul 6179 ce0nn 6180 ce0nnulb 6182