Theorem nfxp 4671

Description: Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfxp	⊢ Ⅎ𝑥(𝐴 × 𝐵)

Proof of Theorem nfxp

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xp 4650	. 2 ⊢ (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
2		nfxp.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2326	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfxp.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2326	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6	3, 5	nfan 1576	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
7	6	nfopab 4086	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
8	1, 7	nfcxfr 2329	1 ⊢ Ⅎ𝑥(𝐴 × 𝐵)

Syntax hints:   ∧ wa 104   ∈ wcel 2160  Ⅎwnfc 2319  {copab 4078   × cxp 4642

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 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-opab 4080  df-xp 4650

This theorem is referenced by:  opeliunxp  4699  nfres  4927  mpomptsx  6222  dmmpossx  6224  fmpox  6225  disjxp1  6261  nfdju  7071  fsum2dlemstep  11474  fisumcom2  11478  fprod2dlemstep  11662  fprodcom2fi  11666