Theorem nfop 3878

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 3678	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2		nfop.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfel1 2385	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
4		nfop.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfel1 2385	. . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V
6	2	nfsn 3729	. . . . . 6 ⊢ Ⅎ𝑥{𝐴}
7	2, 4	nfpr 3719	. . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵}
8	6, 7	nfpr 3719	. . . . 5 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}}
9	8	nfcri 2368	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
10	3, 5, 9	nf3an 1614	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
11	10	nfab 2379	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
12	1, 11	nfcxfr 2371	1 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩

Syntax hints:   ∧ w3a 1004   ∈ wcel 2202  {cab 2217  Ⅎwnfc 2361  Vcvv 2802  {csn 3669  {cpr 3670  ⟨cop 3672

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678

This theorem is referenced by:  nfopd  3879  moop2  4344  fliftfuns  5938  dfmpo  6387  qliftfuns  6787  xpf1o  7029  caucvgprprlemaddq  7927  nfseq  10718  txcnp  14994  cnmpt1t  15008  cnmpt2t  15016