Theorem nfop 4605

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⟩

Proof of Theorem nfop

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 4567	. 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 2670	. . . 4 ⊢ Ⅎx∃w ∈ A z = Phi w
5	4	nfab 2494	. . 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 2670	. . . 4 ⊢ Ⅎx∃w ∈ B z = ( Phi w ∪ {0c})
9	8	nfab 2494	. . 3 ⊢ Ⅎx{z ∣ ∃w ∈ B z = ( Phi w ∪ {0c})}
10	5, 9	nfun 3232	. 2 ⊢ Ⅎx({z ∣ ∃w ∈ A z = Phi w} ∪ {z ∣ ∃w ∈ B z = ( Phi w ∪ {0c})})
11	1, 10	nfcxfr 2487	1 ⊢ Ⅎx⟨A, B⟩

Syntax hints:   = wceq 1642  {cab 2339  Ⅎwnfc 2477  ∃wrex 2616   ∪ cun 3208  {csn 3738  0_cc0c 4375  ⟨cop 4562   Phi cphi 4563

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-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 2479  df-ral 2620  df-rex 2621  df-nin 3212  df-compl 3213  df-un 3215  df-op 4567

This theorem is referenced by:  nfopd  4606