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Theorem dfoprab3 7833
Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
Hypothesis
Ref Expression
dfoprab3.1 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
dfoprab3 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑤   𝑥,𝑧,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem dfoprab3
StepHypRef Expression
1 dfoprab3s 7832 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)}
2 fvex 6739 . . . . 5 (1st𝑤) ∈ V
3 fvex 6739 . . . . 5 (2nd𝑤) ∈ V
4 eqcom 2745 . . . . . . . . . 10 (𝑥 = (1st𝑤) ↔ (1st𝑤) = 𝑥)
5 eqcom 2745 . . . . . . . . . 10 (𝑦 = (2nd𝑤) ↔ (2nd𝑤) = 𝑦)
64, 5anbi12i 630 . . . . . . . . 9 ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) ↔ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦))
7 eqopi 7806 . . . . . . . . 9 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
86, 7sylan2b 597 . . . . . . . 8 ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤))) → 𝑤 = ⟨𝑥, 𝑦⟩)
9 dfoprab3.1 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
108, 9syl 17 . . . . . . 7 ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤))) → (𝜑𝜓))
1110bicomd 226 . . . . . 6 ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤))) → (𝜓𝜑))
1211ex 416 . . . . 5 (𝑤 ∈ (V × V) → ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) → (𝜓𝜑)))
132, 3, 12sbc2iedv 3789 . . . 4 (𝑤 ∈ (V × V) → ([(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓𝜑))
1413pm5.32i 578 . . 3 ((𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
1514opabbii 5129 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
161, 15eqtr2i 2767 1 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  Vcvv 3415  [wsbc 3703  cop 4556  {copab 5124   × cxp 5558  cfv 6389  {coprab 7223  1st c1st 7768  2nd c2nd 7769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pr 5331  ax-un 7532
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-sbc 3704  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-br 5063  df-opab 5125  df-mpt 5145  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-iota 6347  df-fun 6391  df-fv 6397  df-oprab 7226  df-1st 7770  df-2nd 7771
This theorem is referenced by:  dfoprab4  7834  cnvoprab  7839  df1st2  7875  df2nd2  7876
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