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Theorem dfopab2 8040
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 3797 . . . . 5 𝑥[(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑
2119.41 2228 . . . 4 (∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
3 sbcopeq1a 8037 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → ([(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑𝜑))
43pm5.32i 575 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
54exbii 1850 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6 nfcv 2903 . . . . . . . 8 𝑦(1st𝑧)
7 nfsbc1v 3797 . . . . . . . 8 𝑦[(2nd𝑧) / 𝑦]𝜑
86, 7nfsbcw 3799 . . . . . . 7 𝑦[(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑
9819.41 2228 . . . . . 6 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
105, 9bitr3i 276 . . . . 5 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
1110exbii 1850 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
12 elvv 5750 . . . . 5 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
1312anbi1i 624 . . . 4 ((𝑧 ∈ (V × V) ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
142, 11, 133bitr4i 302 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑧 ∈ (V × V) ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑))
1514abbii 2802 . 2 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑)}
16 df-opab 5211 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17 df-rab 3433 . 2 {𝑧 ∈ (V × V) ∣ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑} = {𝑧 ∣ (𝑧 ∈ (V × V) ∧ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑)}
1815, 16, 173eqtr4i 2770 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1st𝑧) / 𝑥][(2nd𝑧) / 𝑦]𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  {crab 3432  Vcvv 3474  [wsbc 3777  cop 4634  {copab 5210   × cxp 5674  cfv 6543  1st c1st 7975  2nd c2nd 7976
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
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-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  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-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7977  df-2nd 7978
This theorem is referenced by: (None)
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