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Theorem iunopabOLD 5560
Description: Obsolete version of iunopab 5559 as of 11-Dec-2024. (Contributed by Stefan O'Rear, 20-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
iunopabOLD 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 𝜑}
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝑧   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑧)

Proof of Theorem iunopabOLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elopab 5527 . . . . 5 (𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
21rexbii 3094 . . . 4 (∃𝑧𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧𝐴𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3 rexcom4 3285 . . . . 5 (∃𝑧𝐴𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑧𝐴𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 rexcom4 3285 . . . . . . 7 (∃𝑧𝐴𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑧𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 r19.42v 3190 . . . . . . . 8 (∃𝑧𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
65exbii 1850 . . . . . . 7 (∃𝑦𝑧𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
74, 6bitri 274 . . . . . 6 (∃𝑧𝐴𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
87exbii 1850 . . . . 5 (∃𝑥𝑧𝐴𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
93, 8bitri 274 . . . 4 (∃𝑧𝐴𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
102, 9bitri 274 . . 3 (∃𝑧𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
1110abbii 2802 . 2 {𝑤 ∣ ∃𝑧𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑)}
12 df-iun 4999 . 2 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}}
13 df-opab 5211 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑)}
1411, 12, 133eqtr4i 2770 1 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wrex 3070  cop 4634   ciun 4997  {copab 5210
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-iun 4999  df-opab 5211
This theorem is referenced by: (None)
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