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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	⊢ ℲxA
nfop.2	⊢ ℲxB

Assertion

Ref	Expression
nfop	⊢ Ⅎx⟨A, B⟩

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Proof of Theorem nfop Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	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 ⊢ ℲxA
3		nfv 1619	. . . . 5 ⊢ Ⅎx z = Phi w
4	2, 3	nfrex 2669	. . . 4 ⊢ Ⅎx∃w ∈ A z = Phi w
5	4	nfab 2493	. . 3 ⊢ Ⅎx{z ∣ ∃w ∈ A z = Phi w}
6		nfop.2	. . . . 5 ⊢ ℲxB
7		nfv 1619	. . . . 5 ⊢ Ⅎx z = ( Phi w ∪ {0c})
8	6, 7	nfrex 2669	. . . 4 ⊢ Ⅎx∃w ∈ B z = ( Phi w ∪ {0c})
9	8	nfab 2493	. . 3 ⊢ Ⅎx{z ∣ ∃w ∈ B z = ( Phi w ∪ {0c})}
10	5, 9	nfun 3231	. 2 ⊢ Ⅎx({z ∣ ∃w ∈ A z = Phi w} ∪ {z ∣ ∃w ∈ B z = ( Phi w ∪ {0c})})
11	1, 10	nfcxfr 2486	1 ⊢ Ⅎx⟨A, 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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