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Theorem nfop 4604
 Description: Bound-variable hypothesis builder for ordered pairs. (Contributed by SF, 2-Jan-2015.)
Hypotheses
Ref Expression
nfop.1 xA
nfop.2 xB
Assertion
Ref Expression
nfop xA, B

Proof of Theorem nfop
Dummy variables z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-op 4566 . 2 A, B = ({z w A z = Phi w} ∪ {z w B z = ( Phi w ∪ {0c})})
2 nfop.1 . . . . 5 xA
3 nfv 1619 . . . . 5 x z = Phi w
42, 3nfrex 2669 . . . 4 xw A z = Phi w
54nfab 2493 . . 3 x{z w A z = Phi w}
6 nfop.2 . . . . 5 xB
7 nfv 1619 . . . . 5 x z = ( Phi w ∪ {0c})
86, 7nfrex 2669 . . . 4 xw B z = ( Phi w ∪ {0c})
98nfab 2493 . . 3 x{z w B z = ( Phi w ∪ {0c})}
105, 9nfun 3231 . 2 x({z w A z = Phi w} ∪ {z w B z = ( Phi w ∪ {0c})})
111, 10nfcxfr 2486 1 xA, B
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  {cab 2339  Ⅎwnfc 2476  ∃wrex 2615   ∪ cun 3207  {csn 3737  0cc0c 4374  ⟨cop 4561   Phi cphi 4562 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-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-nin 3211  df-compl 3212  df-un 3214  df-op 4566 This theorem is referenced by:  nfopd  4605
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