Theorem nfopk 4069

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

Hypotheses

Ref	Expression
nfopk.1	⊢ ℲxA
nfopk.2	⊢ ℲxB

Assertion

Ref	Expression
nfopk	⊢ Ⅎx⟪A, B⟫

Proof of Theorem nfopk

Step	Hyp	Ref	Expression
1		df-opk 4059	. 2 ⊢ ⟪A, B⟫ = {{A}, {A, B}}
2		nfopk.1	. . . 4 ⊢ ℲxA
3	2	nfsn 3785	. . 3 ⊢ Ⅎx{A}
4		nfopk.2	. . . 4 ⊢ ℲxB
5	2, 4	nfpr 3774	. . 3 ⊢ Ⅎx{A, B}
6	3, 5	nfpr 3774	. 2 ⊢ Ⅎx{{A}, {A, B}}
7	1, 6	nfcxfr 2487	1 ⊢ Ⅎx⟪A, B⟫

Syntax hints:  Ⅎwnfc 2477  {csn 3738  {cpr 3739  ⟪copk 4058

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-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by: (None)