Theorem opelopab 2117

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61.

Hypotheses

Ref	Expression
opelopab.1	⊢ A ∈ V
opelopab.2	⊢ B ∈ V
opelopab.3	⊢ (x = A → (φ ↔ ψ))
opelopab.4	⊢ (y = B → (ψ ↔ χ))

Assertion

Ref	Expression
opelopab	⊢ (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ)

Distinct variable group(s):   x,y,A   x,B,y   χ,x,y

Proof of Theorem opelopab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2 ⊢ A ∈ V
2		opelopab.2	. 2 ⊢ B ∈ V
3		opelopab.3	. . 3 ⊢ (x = A → (φ ↔ ψ))
4		opelopab.4	. . 3 ⊢ (y = B → (ψ ↔ χ))
5	3, 4	opelopabg 2115	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ))
6	1, 2, 5	mp2an 520	1 ⊢ (⟨A, B⟩ ∈ {⟨x, y⟩∣φ} ↔ χ)

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810  {copab 2055

This theorem is referenced by:  opelco 2509  fnopabfv 2858  f1oiso 2942  tz7.44-1 2966  tz7.44-2 2967  tz7.44-3 2968  pw2en 3348  tz9.12lem1 3503  tz9.12lem3 3505  aceq3lem 3555  aceq3 3556  infmap2lem1 4951

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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