Theorem opelopab2a 4243

Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)

Hypothesis

Ref	Expression
opelopabga.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
opelopab2a	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opelopab2a

Step	Hyp	Ref	Expression
1		eleq1 2229	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2		eleq1 2229	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷))
3	1, 2	bi2anan9 596	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)))
4		opelopabga.1	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
5	3, 4	anbi12d 465	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑) ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓)))
6	5	opelopabga 4241	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓)))
7	6	bianabs 601	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ⟨cop 3579  {copab 4042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044

This theorem is referenced by:  opelopab2  4248  brab2a  4657  brab2ga  4679  ltdfpr  7447