Theorem nfmpt22 5675

Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Assertion

Ref	Expression
nfmpt22	⊢ Ⅎy(x ∈ A, y ∈ B ↦ C)

Proof of Theorem nfmpt22

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt2 5655	. 2 ⊢ (x ∈ A, y ∈ B ↦ C) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)}
2		nfoprab2 5548	. 2 ⊢ Ⅎy{⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)}
3	1, 2	nfcxfr 2487	1 ⊢ Ⅎy(x ∈ A, y ∈ B ↦ C)

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  {coprab 5528   ↦ cmpt2 5654

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  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-oprab 5529  df-mpt2 5655

This theorem is referenced by:  ov2gf  5712