Theorem nfned 3044

Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfned.1	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfned.2	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfned	⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵)

Proof of Theorem nfned

Step	Hyp	Ref	Expression
1		df-ne 2941	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		nfned.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfned.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfeqd 2916	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
5	4	nfnd 1858	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵)
6	1, 5	nfxfrd 1854	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540  Ⅎwnf 1783  Ⅎwnfc 2890   ≠ wne 2940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-cleq 2729  df-nfc 2892  df-ne 2941

This theorem is referenced by: (None)