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Theorem eloprabga 5578
 Description: The law of concretion for operation class abstraction. Compare elopab 4696. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 18-Jun-2012.) Removed unnecessary distinct variable requirements. (Revised by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
eloprabga.1 ((x = A y = B z = C) → (φψ))
Assertion
Ref Expression
eloprabga ((A V B W C X) → (A, B, C {x, y, z φ} ↔ ψ))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,C,y,z   ψ,x,y,z
Allowed substitution hints:   φ(x,y,z)   V(x,y,z)   W(x,y,z)   X(x,y,z)

Proof of Theorem eloprabga
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2 (A VA V)
2 elex 2867 . 2 (B WB V)
3 elex 2867 . 2 (C XC V)
4 opexg 4587 . . . . 5 ((A V B V) → A, B V)
5 opexg 4587 . . . . 5 ((A, B V C V) → A, B, C V)
64, 5sylan 457 . . . 4 (((A V B V) C V) → A, B, C V)
763impa 1146 . . 3 ((A V B V C V) → A, B, C V)
8 eqeq1 2359 . . . . . . . . . . 11 (w = A, B, C → (w = x, y, zA, B, C = x, y, z))
9 eqcom 2355 . . . . . . . . . . . 12 (A, B, C = x, y, zx, y, z = A, B, C)
10 opth 4602 . . . . . . . . . . . . . 14 (x, y = A, B ↔ (x = A y = B))
1110anbi1i 676 . . . . . . . . . . . . 13 ((x, y = A, B z = C) ↔ ((x = A y = B) z = C))
12 opth 4602 . . . . . . . . . . . . 13 (x, y, z = A, B, C ↔ (x, y = A, B z = C))
13 df-3an 936 . . . . . . . . . . . . 13 ((x = A y = B z = C) ↔ ((x = A y = B) z = C))
1411, 12, 133bitr4i 268 . . . . . . . . . . . 12 (x, y, z = A, B, C ↔ (x = A y = B z = C))
159, 14bitri 240 . . . . . . . . . . 11 (A, B, C = x, y, z ↔ (x = A y = B z = C))
168, 15syl6bb 252 . . . . . . . . . 10 (w = A, B, C → (w = x, y, z ↔ (x = A y = B z = C)))
1716anbi1d 685 . . . . . . . . 9 (w = A, B, C → ((w = x, y, z φ) ↔ ((x = A y = B z = C) φ)))
18 eloprabga.1 . . . . . . . . . 10 ((x = A y = B z = C) → (φψ))
1918pm5.32i 618 . . . . . . . . 9 (((x = A y = B z = C) φ) ↔ ((x = A y = B z = C) ψ))
2017, 19syl6bb 252 . . . . . . . 8 (w = A, B, C → ((w = x, y, z φ) ↔ ((x = A y = B z = C) ψ)))
21203exbidv 1629 . . . . . . 7 (w = A, B, C → (xyz(w = x, y, z φ) ↔ xyz((x = A y = B z = C) ψ)))
2221adantl 452 . . . . . 6 (((A V B V C V) w = A, B, C) → (xyz(w = x, y, z φ) ↔ xyz((x = A y = B z = C) ψ)))
23 df-oprab 5528 . . . . . . . . . 10 {x, y, z φ} = {w xyz(w = x, y, z φ)}
2423eleq2i 2417 . . . . . . . . 9 (w {x, y, z φ} ↔ w {w xyz(w = x, y, z φ)})
25 abid 2341 . . . . . . . . 9 (w {w xyz(w = x, y, z φ)} ↔ xyz(w = x, y, z φ))
2624, 25bitr2i 241 . . . . . . . 8 (xyz(w = x, y, z φ) ↔ w {x, y, z φ})
27 eleq1 2413 . . . . . . . 8 (w = A, B, C → (w {x, y, z φ} ↔ A, B, C {x, y, z φ}))
2826, 27syl5bb 248 . . . . . . 7 (w = A, B, C → (xyz(w = x, y, z φ) ↔ A, B, C {x, y, z φ}))
2928adantl 452 . . . . . 6 (((A V B V C V) w = A, B, C) → (xyz(w = x, y, z φ) ↔ A, B, C {x, y, z φ}))
30 isset 2863 . . . . . . . . . . . 12 (A V ↔ x x = A)
31 isset 2863 . . . . . . . . . . . 12 (B V ↔ y y = B)
32 isset 2863 . . . . . . . . . . . 12 (C V ↔ z z = C)
3330, 31, 323anbi123i 1140 . . . . . . . . . . 11 ((A V B V C V) ↔ (x x = A y y = B z z = C))
34 eeeanv 1914 . . . . . . . . . . 11 (xyz(x = A y = B z = C) ↔ (x x = A y y = B z z = C))
3533, 34bitr4i 243 . . . . . . . . . 10 ((A V B V C V) ↔ xyz(x = A y = B z = C))
3635biimpi 186 . . . . . . . . 9 ((A V B V C V) → xyz(x = A y = B z = C))
3736biantrurd 494 . . . . . . . 8 ((A V B V C V) → (ψ ↔ (xyz(x = A y = B z = C) ψ)))
38 19.41vvv 1903 . . . . . . . 8 (xyz((x = A y = B z = C) ψ) ↔ (xyz(x = A y = B z = C) ψ))
3937, 38syl6rbbr 255 . . . . . . 7 ((A V B V C V) → (xyz((x = A y = B z = C) ψ) ↔ ψ))
4039adantr 451 . . . . . 6 (((A V B V C V) w = A, B, C) → (xyz((x = A y = B z = C) ψ) ↔ ψ))
4122, 29, 403bitr3d 274 . . . . 5 (((A V B V C V) w = A, B, C) → (A, B, C {x, y, z φ} ↔ ψ))
4241expcom 424 . . . 4 (w = A, B, C → ((A V B V C V) → (A, B, C {x, y, z φ} ↔ ψ)))
4342vtocleg 2925 . . 3 (A, B, C V → ((A V B V C V) → (A, B, C {x, y, z φ} ↔ ψ)))
447, 43mpcom 32 . 2 ((A V B V C V) → (A, B, C {x, y, z φ} ↔ ψ))
451, 2, 3, 44syl3an 1224 1 ((A V B W C X) → (A, B, C {x, y, z φ} ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ⟨cop 4561  {coprab 5527 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-oprab 5528 This theorem is referenced by:  eloprabg  5579  ovigg  5596
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