Theorem nfxp 4811

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 4785	. 2 ⊢ (A × B) = {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B)}
2		nfxp.1	. . . . 5 ⊢ ℲxA
3	2	nfcri 2484	. . . 4 ⊢ Ⅎx y ∈ A
4		nfxp.2	. . . . 5 ⊢ ℲxB
5	4	nfcri 2484	. . . 4 ⊢ Ⅎx z ∈ B
6	3, 5	nfan 1824	. . 3 ⊢ Ⅎx(y ∈ A ∧ z ∈ B)
7	6	nfopab 4628	. 2 ⊢ Ⅎx{⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B)}
8	1, 7	nfcxfr 2487	1 ⊢ Ⅎx(A × B)

This proof depends on syntax axioms:   ∧ wa 358   ∈ wcel 1710  Ⅎwnfc 2477  {copab 4623   × cxp 4771

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-opab 4624  df-xp 4785

This theorem is used by:  opeliunxp  4821  nfres  4937  fmpt2x  5731