Theorem nfop 3824

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 3631	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2		nfop.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfel1 2350	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
4		nfop.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfel1 2350	. . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V
6	2	nfsn 3682	. . . . . 6 ⊢ Ⅎ𝑥{𝐴}
7	2, 4	nfpr 3672	. . . . . 6 ⊢ Ⅎ𝑥{𝐴, 𝐵}
8	6, 7	nfpr 3672	. . . . 5 ⊢ Ⅎ𝑥{{𝐴}, {𝐴, 𝐵}}
9	8	nfcri 2333	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
10	3, 5, 9	nf3an 1580	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
11	10	nfab 2344	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
12	1, 11	nfcxfr 2336	1 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩

Syntax hints:   ∧ w3a 980   ∈ wcel 2167  {cab 2182  Ⅎwnfc 2326  Vcvv 2763  {csn 3622  {cpr 3623  ⟨cop 3625

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631

This theorem is referenced by:  nfopd  3825  moop2  4284  fliftfuns  5845  dfmpo  6281  qliftfuns  6678  xpf1o  6905  caucvgprprlemaddq  7775  nfseq  10549  txcnp  14507  cnmpt1t  14521  cnmpt2t  14529