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Theorem iunopabOLD 5518
Description: Obsolete version of iunopab 5517 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 5485 . . . . 5 (𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
21rexbii 3098 . . . 4 (∃𝑧𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧𝐴𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3 rexcom4 3272 . . . . 5 (∃𝑧𝐴𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑧𝐴𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 rexcom4 3272 . . . . . . 7 (∃𝑧𝐴𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑧𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 r19.42v 3188 . . . . . . . 8 (∃𝑧𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
65exbii 1851 . . . . . . 7 (∃𝑦𝑧𝐴 (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
74, 6bitri 275 . . . . . 6 (∃𝑧𝐴𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
87exbii 1851 . . . . 5 (∃𝑥𝑧𝐴𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
93, 8bitri 275 . . . 4 (∃𝑧𝐴𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
102, 9bitri 275 . . 3 (∃𝑧𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑))
1110abbii 2807 . 2 {𝑤 ∣ ∃𝑧𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑)}
12 df-iun 4957 . 2 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧𝐴 𝑤 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}}
13 df-opab 5169 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝐴 𝜑)}
1411, 12, 133eqtr4i 2775 1 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2714  wrex 3074  cop 4593   ciun 4955  {copab 5168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rex 3075  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-iun 4957  df-opab 5169
This theorem is referenced by: (None)
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