Theorem opkelopkab 4247

Description: Kuratowski ordered pair membership in an abstraction of Kuratowski ordered pairs. (Contributed by SF, 12-Jan-2015.)

Hypotheses

Ref	Expression
opkelopkab.1	⊢ A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)}
opkelopkab.2	⊢ (y = B → (φ ↔ ψ))
opkelopkab.3	⊢ (z = C → (ψ ↔ χ))
opkelopkab.4	⊢ B ∈ V
opkelopkab.5	⊢ C ∈ V

Assertion

Ref	Expression
opkelopkab	⊢ (⟪B, C⟫ ∈ A ↔ χ)

Distinct variable groups:   y,A,z   x,B,y,z   x,C,y,z   χ,z   φ,x   ψ,y   x,y,z

Allowed substitution hints:   φ(y,z)   ψ(x,z)   χ(x,y)   A(x)

Proof of Theorem opkelopkab

Step	Hyp	Ref	Expression
1		opkelopkab.4	. 2 ⊢ B ∈ V
2		opkelopkab.5	. 2 ⊢ C ∈ V
3		opkelopkab.1	. . 3 ⊢ A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)}
4		opkelopkab.2	. . 3 ⊢ (y = B → (φ ↔ ψ))
5		opkelopkab.3	. . 3 ⊢ (z = C → (ψ ↔ χ))
6	3, 4, 5	opkelopkabg 4246	. 2 ⊢ ((B ∈ V ∧ C ∈ V) → (⟪B, C⟫ ∈ A ↔ χ))
7	1, 2, 6	mp2an 653	1 ⊢ (⟪B, C⟫ ∈ A ↔ χ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860  ⟪copk 4058

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  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by:  sikss1c1c  4268  dfima2  4746  dfco1  4749  dfsi2  4752