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Theorem nfrex 2669
 Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfrex.1 xA
nfrex.2 xφ
Assertion
Ref Expression
nfrex xy A φ

Proof of Theorem nfrex
StepHypRef Expression
1 dfrex2 2627 . 2 (y A φ ↔ ¬ y A ¬ φ)
2 nfrex.1 . . . 4 xA
3 nfrex.2 . . . . 5 xφ
43nfn 1793 . . . 4 x ¬ φ
52, 4nfral 2667 . . 3 xy A ¬ φ
65nfn 1793 . 2 x ¬ y A ¬ φ
71, 6nfxfr 1570 1 xy A φ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  Ⅎwnf 1544  Ⅎwnfc 2476  ∀wral 2614  ∃wrex 2615 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620 This theorem is referenced by:  r19.12  2727  sbcrexg  3121  nfuni  3897  nfiun  3995  nfop  4604  nfima  4953
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