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Theorem nfned 2612
 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 (φxA)
nfned.2 (φxB)
Assertion
Ref Expression
nfned (φ → Ⅎx AB)

Proof of Theorem nfned
StepHypRef Expression
1 df-ne 2518 . 2 (AB ↔ ¬ A = B)
2 nfned.1 . . . 4 (φxA)
3 nfned.2 . . . 4 (φxB)
42, 3nfeqd 2503 . . 3 (φ → Ⅎx A = B)
54nfnd 1791 . 2 (φ → Ⅎx ¬ A = B)
61, 5nfxfrd 1571 1 (φ → Ⅎx AB)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1544   = wceq 1642  Ⅎwnfc 2476   ≠ wne 2516 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 2478  df-ne 2518 This theorem is referenced by: (None)
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