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Theorem eloprabi 5764
Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabi.1 (x = (1st ‘(1stA)) → (φψ))
eloprabi.2 (y = (2nd ‘(1stA)) → (ψχ))
eloprabi.3 (z = (2ndA) → (χθ))
Assertion
Ref Expression
eloprabi (A {⟨⟨x, y⟩, z⟩ ∣ φ} → θ)
Distinct variable groups:   x,y,z,A   θ,x,y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)   χ(x,y,z)

Proof of Theorem eloprabi
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . . . . 6 (w = A → (w = ⟨⟨x, y⟩, z⟩ ↔ A = ⟨⟨x, y⟩, z⟩))
21anbi1d 438 . . . . 5 (w = A → ((w = ⟨⟨x, y⟩, z φ) ↔ (A = ⟨⟨x, y⟩, z φ)))
323exbidv 1746 . . . 4 (w = A → (xyz(w = ⟨⟨x, y⟩, z φ) ↔ xyz(A = ⟨⟨x, y⟩, z φ)))
4 df-oprab 5459 . . . 4 {⟨⟨x, y⟩, z⟩ ∣ φ} = {wxyz(w = ⟨⟨x, y⟩, z φ)}
53, 4elab2g 2683 . . 3 (A {⟨⟨x, y⟩, z⟩ ∣ φ} → (A {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ xyz(A = ⟨⟨x, y⟩, z φ)))
65ibi 165 . 2 (A {⟨⟨x, y⟩, z⟩ ∣ φ} → xyz(A = ⟨⟨x, y⟩, z φ))
7 vex 2554 . . . . . . . . . . . 12 x V
8 vex 2554 . . . . . . . . . . . 12 y V
97, 8opex 3957 . . . . . . . . . . 11 x, y V
10 vex 2554 . . . . . . . . . . 11 z V
119, 10op1std 5717 . . . . . . . . . 10 (A = ⟨⟨x, y⟩, z⟩ → (1stA) = ⟨x, y⟩)
1211fveq2d 5125 . . . . . . . . 9 (A = ⟨⟨x, y⟩, z⟩ → (1st ‘(1stA)) = (1st ‘⟨x, y⟩))
137, 8op1st 5715 . . . . . . . . 9 (1st ‘⟨x, y⟩) = x
1412, 13syl6req 2086 . . . . . . . 8 (A = ⟨⟨x, y⟩, z⟩ → x = (1st ‘(1stA)))
15 eloprabi.1 . . . . . . . 8 (x = (1st ‘(1stA)) → (φψ))
1614, 15syl 14 . . . . . . 7 (A = ⟨⟨x, y⟩, z⟩ → (φψ))
1711fveq2d 5125 . . . . . . . . 9 (A = ⟨⟨x, y⟩, z⟩ → (2nd ‘(1stA)) = (2nd ‘⟨x, y⟩))
187, 8op2nd 5716 . . . . . . . . 9 (2nd ‘⟨x, y⟩) = y
1917, 18syl6req 2086 . . . . . . . 8 (A = ⟨⟨x, y⟩, z⟩ → y = (2nd ‘(1stA)))
20 eloprabi.2 . . . . . . . 8 (y = (2nd ‘(1stA)) → (ψχ))
2119, 20syl 14 . . . . . . 7 (A = ⟨⟨x, y⟩, z⟩ → (ψχ))
229, 10op2ndd 5718 . . . . . . . . 9 (A = ⟨⟨x, y⟩, z⟩ → (2ndA) = z)
2322eqcomd 2042 . . . . . . . 8 (A = ⟨⟨x, y⟩, z⟩ → z = (2ndA))
24 eloprabi.3 . . . . . . . 8 (z = (2ndA) → (χθ))
2523, 24syl 14 . . . . . . 7 (A = ⟨⟨x, y⟩, z⟩ → (χθ))
2616, 21, 253bitrd 203 . . . . . 6 (A = ⟨⟨x, y⟩, z⟩ → (φθ))
2726biimpa 280 . . . . 5 ((A = ⟨⟨x, y⟩, z φ) → θ)
2827exlimiv 1486 . . . 4 (z(A = ⟨⟨x, y⟩, z φ) → θ)
2928exlimiv 1486 . . 3 (yz(A = ⟨⟨x, y⟩, z φ) → θ)
3029exlimiv 1486 . 2 (xyz(A = ⟨⟨x, y⟩, z φ) → θ)
316, 30syl 14 1 (A {⟨⟨x, y⟩, z⟩ ∣ φ} → θ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  cop 3370  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-fv 4853  df-oprab 5459  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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