Theorem nfxp 5692

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 5665	. 2 ⊢ (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
2		nfxp.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2923	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfxp.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2923	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6	3, 5	nfan 1926	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
7	6	nfopab 5181	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
8	1, 7	nfcxfr 2929	1 ⊢ Ⅎ𝑥(𝐴 × 𝐵)

Syntax hints:   ∧ wa 400   ∈ wcel 2149  Ⅎwnfc 2916  {copab 5174   × cxp 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-opab 5175  df-xp 5665

This theorem is referenced by:  opeliunxp  5726  opeliun2xp  5727  nfres  5978  mpomptsx  8057  dmmpossx  8059  fmpox  8060  ovmptss  8084  nfdju  9889  axcc2  10417  fsum2dlem  15817  fsumcom2  15821  fprod2dlem  16030  fprodcom2  16034  gsumcom2  20041  prdsdsf  24489  prdsxmet  24491  iunxpssiun1  32850  djussxp2  32930  aciunf1lem  32944  gsumpart  33320  esum2dlem  34423  poimirlem16  38170  poimirlem19  38173  dvnprodlem1  46547  stoweidlem21  46622  stoweidlem47  46648  dmmpossx2  48997