Theorem nfxp 5688

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 5661	. 2 ⊢ (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
2		nfxp.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2914	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfxp.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2914	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6	3, 5	nfan 1932	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
7	6	nfopab 5174	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
8	1, 7	nfcxfr 2920	1 ⊢ Ⅎ𝑥(𝐴 × 𝐵)

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2145  Ⅎwnfc 2907  {copab 5167   × cxp 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-opab 5168  df-xp 5661

This theorem is used by:  opeliunxp  5722  opeliun2xp  5723  nfres  5974  mpomptsx  8061  dmmpossx  8063  fmpox  8064  ovmptss  8090  nfdju  9912  axcc2  10439  fsum2dlem  15856  fsumcom2  15860  fprod2dlem  16067  fprodcom2  16071  gsumcom2  20102  prdsdsf  24593  prdsxmet  24595  iunxpssiun1  33041  djussxp2  33121  aciunf1lem  33135  gsumpart  33503  esum2dlem  34602  poimirlem16  38385  poimirlem19  38388  dvnprodlem1  46774  stoweidlem21  46849  stoweidlem47  46875  dmmpossx2  49267