Theorem nfop 3876

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

Hypotheses

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

Assertion

Ref	Expression
nfop	⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩

Proof of Theorem nfop

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 3676	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2		nfop.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfel1 2383	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
4		nfop.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfel1 2383	. . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V
6	2	nfsn 3727	. . . . . 6 ⊢ Ⅎ𝑥{𝐴}
7	2, 4	nfpr 3717	. . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵}
8	6, 7	nfpr 3717	. . . . 5 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}}
9	8	nfcri 2366	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
10	3, 5, 9	nf3an 1612	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
11	10	nfab 2377	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
12	1, 11	nfcxfr 2369	1 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩

Syntax hints:   ∧ w3a 1002   ∈ wcel 2200  {cab 2215  Ⅎwnfc 2359  Vcvv 2800  {csn 3667  {cpr 3668  ⟨cop 3670

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676

This theorem is referenced by:  nfopd  3877  moop2  4342  fliftfuns  5934  dfmpo  6383  qliftfuns  6783  xpf1o  7025  caucvgprprlemaddq  7918  nfseq  10709  txcnp  14985  cnmpt1t  14999  cnmpt2t  15007