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Theorem opelopab3 37725
Description: Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)
Hypotheses
Ref Expression
opelopab3.1 (𝑥 = 𝐴 → (𝜑𝜓))
opelopab3.2 (𝑦 = 𝐵 → (𝜓𝜒))
opelopab3.3 (𝜒𝐴𝐶)
Assertion
Ref Expression
opelopab3 (𝐵𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem opelopab3
StepHypRef Expression
1 elopaelxp 5775 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2 opelxp1 5727 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (V × V) → 𝐴 ∈ V)
31, 2syl 17 . . . 4 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
43anim1i 615 . . 3 ((⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐵𝐷) → (𝐴 ∈ V ∧ 𝐵𝐷))
54ancoms 458 . 2 ((𝐵𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵𝐷))
6 opelopab3.3 . . . . 5 (𝜒𝐴𝐶)
76elexd 3504 . . . 4 (𝜒𝐴 ∈ V)
87anim1i 615 . . 3 ((𝜒𝐵𝐷) → (𝐴 ∈ V ∧ 𝐵𝐷))
98ancoms 458 . 2 ((𝐵𝐷𝜒) → (𝐴 ∈ V ∧ 𝐵𝐷))
10 opelopab3.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
11 opelopab3.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
1210, 11opelopabg 5543 . 2 ((𝐴 ∈ V ∧ 𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
135, 9, 12pm5.21nd 802 1 (𝐵𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  {copab 5205   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-xp 5691
This theorem is referenced by: (None)
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