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Theorem dfoprab3 7727
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 7726 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)}
2 fvex 6656 . . . . 5 (1st𝑤) ∈ V
3 fvex 6656 . . . . 5 (2nd𝑤) ∈ V
4 eqcom 2828 . . . . . . . . . 10 (𝑥 = (1st𝑤) ↔ (1st𝑤) = 𝑥)
5 eqcom 2828 . . . . . . . . . 10 (𝑦 = (2nd𝑤) ↔ (2nd𝑤) = 𝑦)
64, 5anbi12i 629 . . . . . . . . 9 ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) ↔ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦))
7 eqopi 7700 . . . . . . . . 9 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
86, 7sylan2b 596 . . . . . . . 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 3826 . . . 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 5106 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
161, 15eqtr2i 2845 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 1538  wcel 2115  Vcvv 3471  [wsbc 3749  cop 4546  {copab 5101   × cxp 5526  cfv 6328  {coprab 7131  1st c1st 7662  2nd c2nd 7663
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fv 6336  df-oprab 7134  df-1st 7664  df-2nd 7665
This theorem is referenced by:  dfoprab4  7728  cnvoprab  7733  df1st2  7768  df2nd2  7769
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