Theorem nfned 2434

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 2341	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		nfned.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfned.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfeqd 2327	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
5	4	nfnd 1650	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵)
6	1, 5	nfxfrd 1468	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1348  Ⅎwnf 1453  Ⅎwnfc 2299   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-cleq 2163  df-nfc 2301  df-ne 2341

This theorem is referenced by: (None)