Theorem nfned 2613

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 2519	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
2		nfned.1	. . . 4 ⊢ (φ → ℲxA)
3		nfned.2	. . . 4 ⊢ (φ → ℲxB)
4	2, 3	nfeqd 2504	. . 3 ⊢ (φ → Ⅎx A = B)
5	4	nfnd 1791	. 2 ⊢ (φ → Ⅎx ¬ A = B)
6	1, 5	nfxfrd 1571	1 ⊢ (φ → Ⅎx A ≠ B)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1544   = wceq 1642  Ⅎwnfc 2477   ≠ wne 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-nfc 2479  df-ne 2519

This theorem is referenced by: (None)