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Theorem opelopab3 33842
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 5348 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2 opelxp1 5307 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (V × V) → 𝐴 ∈ V)
31, 2syl 17 . . . 4 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
43anim1i 593 . . 3 ((⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐵𝐷) → (𝐴 ∈ V ∧ 𝐵𝐷))
54ancoms 468 . 2 ((𝐵𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵𝐷))
6 opelopab3.3 . . . . 5 (𝜒𝐴𝐶)
7 elex 3352 . . . . 5 (𝐴𝐶𝐴 ∈ V)
86, 7syl 17 . . . 4 (𝜒𝐴 ∈ V)
98anim1i 593 . . 3 ((𝜒𝐵𝐷) → (𝐴 ∈ V ∧ 𝐵𝐷))
109ancoms 468 . 2 ((𝐵𝐷𝜒) → (𝐴 ∈ V ∧ 𝐵𝐷))
11 opelopab3.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
12 opelopab3.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
1311, 12opelopabg 5143 . 2 ((𝐴 ∈ V ∧ 𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
145, 10, 13pm5.21nd 979 1 (𝐵𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327  {copab 4864   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272
This theorem is referenced by: (None)
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