Theorem nfop 4605

Description: Bound-variable hypothesis builder for ordered pairs. (Contributed by SF, 2-Jan-2015.)

Hypotheses

Ref	Expression
nfop.1	|- F/_xA
nfop.2	|- F/_xB

Assertion

Ref	Expression
nfop	|- F/_x<.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 | E.w e. A z = Phi w} u. {z | E.w e. B z = ( Phi w u. {0c})})
2		nfop.1	. . . . 5 |- F/_xA
3		nfv 1619	. . . . 5 |- F/x z = Phi w
4	2, 3	nfrex 2670	. . . 4 |- F/xE.w e. A z = Phi w
5	4	nfab 2494	. . 3 |- F/_x{z | E.w e. A z = Phi w}
6		nfop.2	. . . . 5 |- F/_xB
7		nfv 1619	. . . . 5 |- F/x z = ( Phi w u. {0c})
8	6, 7	nfrex 2670	. . . 4 |- F/xE.w e. B z = ( Phi w u. {0c})
9	8	nfab 2494	. . 3 |- F/_x{z | E.w e. B z = ( Phi w u. {0c})}
10	5, 9	nfun 3232	. 2 |- F/_x({z | E.w e. A z = Phi w} u. {z | E.w e. B z = ( Phi w u. {0c})})
11	1, 10	nfcxfr 2487	1 |- F/_x<.A, B>.

Syntax hints:   = wceq 1642  {cab 2339   F/_wnfc 2477  E.wrex 2616   u. 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