Theorem nfxp 5651

Description: Bound-variable hypothesis builder for Cartesian 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 5624	. 2 ⊢ (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
2		nfxp.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2893	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfxp.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2893	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6	3, 5	nfan 1906	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
7	6	nfopab 5141	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
8	1, 7	nfcxfr 2899	1 ⊢ Ⅎ𝑥(𝐴 × 𝐵)

Syntax hints:   ∧ wa 396   ∈ wcel 2119  Ⅎwnfc 2886  {copab 5134   × cxp 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-opab 5135  df-xp 5624

This theorem is referenced by:  opeliunxp  5685  opeliun2xp  5686  nfres  5933  mpomptsx  8006  dmmpossx  8008  fmpox  8009  ovmptss  8032  nfdju  9822  axcc2  10350  fsum2dlem  15723  fsumcom2  15727  fprod2dlem  15936  fprodcom2  15940  gsumcom2  19941  prdsdsf  24350  prdsxmet  24352  iunxpssiun1  32657  djussxp2  32740  aciunf1lem  32754  gsumpart  33144  esum2dlem  34276  poimirlem16  38003  poimirlem19  38006  dvnprodlem1  46389  stoweidlem21  46464  stoweidlem47  46490  dmmpossx2  48828