Theorem nfop 3806

Description: Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)

Hypotheses

Ref	Expression
nfop.1	|- F/_ x A
nfop.2	|- F/_ x B

Assertion

Ref	Expression
nfop	|- F/_ x <. A , B >.

Proof of Theorem nfop

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 3613	. 2 |- <. A , B >. = { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
2		nfop.1	. . . . 5 |- F/_ x A
3	2	nfel1 2340	. . . 4 |- F/ x A e. _V
4		nfop.2	. . . . 5 |- F/_ x B
5	4	nfel1 2340	. . . 4 |- F/ x B e. _V
6	2	nfsn 3664	. . . . . 6 |- F/_ x { A }
7	2, 4	nfpr 3654	. . . . . 6 |- F/_ x { A , B }
8	6, 7	nfpr 3654	. . . . 5 |- F/_ x { { A } , { A , B } }
9	8	nfcri 2323	. . . 4 |- F/ x y e. { { A } , { A , B } }
10	3, 5, 9	nf3an 1576	. . 3 |- F/ x ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } )
11	10	nfab 2334	. 2 |- F/_ x { y | ( A e. _V /\ B e. _V /\ y e. { { A } , { A , B } } ) }
12	1, 11	nfcxfr 2326	1 |- F/_ x <. A , B >.

Syntax hints:   /\ w3a 979   e. wcel 2158   {cab 2173   F/_wnfc 2316   _Vcvv 2749   {csn 3604   {cpr 3605   <.cop 3607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613

This theorem is referenced by:  nfopd  3807  moop2  4263  fliftfuns  5812  dfmpo  6237  qliftfuns  6632  xpf1o  6857  caucvgprprlemaddq  7720  nfseq  10468  txcnp  14042  cnmpt1t  14056  cnmpt2t  14064