Theorem nfop 3820

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 3627	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2		nfop.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfel1 2347	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
4		nfop.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfel1 2347	. . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V
6	2	nfsn 3678	. . . . . 6 ⊢ Ⅎ𝑥{𝐴}
7	2, 4	nfpr 3668	. . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵}
8	6, 7	nfpr 3668	. . . . 5 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}}
9	8	nfcri 2330	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
10	3, 5, 9	nf3an 1577	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
11	10	nfab 2341	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
12	1, 11	nfcxfr 2333	1 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩

Syntax hints:   ∧ w3a 980   ∈ wcel 2164  {cab 2179  Ⅎwnfc 2323  Vcvv 2760  {csn 3618  {cpr 3619  ⟨cop 3621

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627

This theorem is referenced by:  nfopd  3821  moop2  4280  fliftfuns  5841  dfmpo  6276  qliftfuns  6673  xpf1o  6900  caucvgprprlemaddq  7768  nfseq  10528  txcnp  14439  cnmpt1t  14453  cnmpt2t  14461