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Theorem dfopab2 7985
Description: A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfopab2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑}
Distinct variable groups:   𝜑,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dfopab2
StepHypRef Expression
1 nfsbc1v 3760 . . . . 5 𝑥[(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑
2119.41 2229 . . . 4 (∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
3 sbcopeq1a 7982 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → ([(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑𝜑))
43pm5.32i 576 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
54exbii 1851 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6 nfcv 2908 . . . . . . . 8 𝑦(1st𝑧)
7 nfsbc1v 3760 . . . . . . . 8 𝑦[(2nd𝑧) / 𝑦]𝜑
86, 7nfsbcw 3762 . . . . . . 7 𝑦[(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑
9819.41 2229 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
105, 9bitr3i 277 . . . . 5 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
1110exbii 1851 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
12 elvv 5707 . . . . 5 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
1312anbi1i 625 . . . 4 ((𝑧 ∈ (V × V) ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
142, 11, 133bitr4i 303 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑧 ∈ (V × V) ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
1514abbii 2807 . 2 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑)}
16 df-opab 5169 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17 df-rab 3409 . 2 {𝑧 ∈ (V × V) ∣ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑)}
1815, 16, 173eqtr4i 2775 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2714  {crab 3408  Vcvv 3446  [wsbc 3740  cop 4593  {copab 5168   × cxp 5632  cfv 6497  1st c1st 7920  2nd c2nd 7921
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  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-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by: (None)
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