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Theorem opelopab3 35117
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 5618 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2 opelxp1 5573 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (V × V) → 𝐴 ∈ V)
31, 2syl 17 . . . 4 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
43anim1i 617 . . 3 ((⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐵𝐷) → (𝐴 ∈ V ∧ 𝐵𝐷))
54ancoms 462 . 2 ((𝐵𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵𝐷))
6 opelopab3.3 . . . . 5 (𝜒𝐴𝐶)
76elexd 3489 . . . 4 (𝜒𝐴 ∈ V)
87anim1i 617 . . 3 ((𝜒𝐵𝐷) → (𝐴 ∈ V ∧ 𝐵𝐷))
98ancoms 462 . 2 ((𝐵𝐷𝜒) → (𝐴 ∈ V ∧ 𝐵𝐷))
10 opelopab3.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
11 opelopab3.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
1210, 11opelopabg 5402 . 2 ((𝐴 ∈ V ∧ 𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
135, 9, 12pm5.21nd 801 1 (𝐵𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  cop 4545  {copab 5104   × cxp 5530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-opab 5105  df-xp 5538
This theorem is referenced by: (None)
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