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Theorem dfoprab3 5759
 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 5758 . 2 {⟨⟨x, y⟩, z⟩ ∣ ψ} = {⟨w, z⟩ ∣ (w (V × V) [(1stw) / x][(2ndw) / y]ψ)}
2 vex 2554 . . . . . 6 w V
3 1stexg 5736 . . . . . 6 (w V → (1stw) V)
42, 3ax-mp 7 . . . . 5 (1stw) V
5 2ndexg 5737 . . . . . 6 (w V → (2ndw) V)
62, 5ax-mp 7 . . . . 5 (2ndw) V
7 eqcom 2039 . . . . . . . . . 10 (x = (1stw) ↔ (1stw) = x)
8 eqcom 2039 . . . . . . . . . 10 (y = (2ndw) ↔ (2ndw) = y)
97, 8anbi12i 433 . . . . . . . . 9 ((x = (1stw) y = (2ndw)) ↔ ((1stw) = x (2ndw) = y))
10 eqopi 5740 . . . . . . . . 9 ((w (V × V) ((1stw) = x (2ndw) = y)) → w = ⟨x, y⟩)
119, 10sylan2b 271 . . . . . . . 8 ((w (V × V) (x = (1stw) y = (2ndw))) → w = ⟨x, y⟩)
12 dfoprab3.1 . . . . . . . 8 (w = ⟨x, y⟩ → (φψ))
1311, 12syl 14 . . . . . . 7 ((w (V × V) (x = (1stw) y = (2ndw))) → (φψ))
1413bicomd 129 . . . . . 6 ((w (V × V) (x = (1stw) y = (2ndw))) → (ψφ))
1514ex 108 . . . . 5 (w (V × V) → ((x = (1stw) y = (2ndw)) → (ψφ)))
164, 6, 15sbc2iedv 2824 . . . 4 (w (V × V) → ([(1stw) / x][(2ndw) / y]ψφ))
1716pm5.32i 427 . . 3 ((w (V × V) [(1stw) / x][(2ndw) / y]ψ) ↔ (w (V × V) φ))
1817opabbii 3815 . 2 {⟨w, z⟩ ∣ (w (V × V) [(1stw) / x][(2ndw) / y]ψ)} = {⟨w, z⟩ ∣ (w (V × V) φ)}
191, 18eqtr2i 2058 1 {⟨w, z⟩ ∣ (w (V × V) φ)} = {⟨⟨x, y⟩, z⟩ ∣ ψ}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551  [wsbc 2758  ⟨cop 3370  {copab 3808   × cxp 4286  ‘cfv 4845  {coprab 5456  1st c1st 5707  2nd c2nd 5708 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-oprab 5459  df-1st 5709  df-2nd 5710 This theorem is referenced by:  dfoprab4  5760  df1st2  5782  df2nd2  5783
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