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Theorem eloprabga 5514
Description: The law of concretion for operation class abstraction. Compare elopab 3969. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (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 2543 . 2 (A 𝑉A V)
2 elex 2543 . 2 (B 𝑊B V)
3 elex 2543 . 2 (𝐶 𝑋𝐶 V)
4 opexg 3938 . . . . 5 ((A V B V) → ⟨A, B V)
5 opexg 3938 . . . . 5 ((⟨A, B V 𝐶 V) → ⟨⟨A, B⟩, 𝐶 V)
64, 5sylan 267 . . . 4 (((A V B V) 𝐶 V) → ⟨⟨A, B⟩, 𝐶 V)
763impa 1085 . . 3 ((A V B V 𝐶 V) → ⟨⟨A, B⟩, 𝐶 V)
8 simpr 103 . . . . . . . . . . 11 (((A V B V 𝐶 V) w = ⟨⟨A, B⟩, 𝐶⟩) → w = ⟨⟨A, B⟩, 𝐶⟩)
98eqeq1d 2030 . . . . . . . . . 10 (((A V B V 𝐶 V) w = ⟨⟨A, B⟩, 𝐶⟩) → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨A, B⟩, 𝐶⟩ = ⟨⟨x, y⟩, z⟩))
10 eqcom 2024 . . . . . . . . . . 11 (⟨⟨A, B⟩, 𝐶⟩ = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, 𝐶⟩)
11 vex 2538 . . . . . . . . . . . 12 x V
12 vex 2538 . . . . . . . . . . . 12 y V
13 vex 2538 . . . . . . . . . . . 12 z V
1411, 12, 13otth2 3952 . . . . . . . . . . 11 (⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, 𝐶⟩ ↔ (x = A y = B z = 𝐶))
1510, 14bitri 173 . . . . . . . . . 10 (⟨⟨A, B⟩, 𝐶⟩ = ⟨⟨x, y⟩, z⟩ ↔ (x = A y = B z = 𝐶))
169, 15syl6bb 185 . . . . . . . . 9 (((A V B V 𝐶 V) w = ⟨⟨A, B⟩, 𝐶⟩) → (w = ⟨⟨x, y⟩, z⟩ ↔ (x = A y = B z = 𝐶)))
1716anbi1d 441 . . . . . . . 8 (((A V B V 𝐶 V) w = ⟨⟨A, B⟩, 𝐶⟩) → ((w = ⟨⟨x, y⟩, z φ) ↔ ((x = A y = B z = 𝐶) φ)))
18 eloprabga.1 . . . . . . . . 9 ((x = A y = B z = 𝐶) → (φψ))
1918pm5.32i 430 . . . . . . . 8 (((x = A y = B z = 𝐶) φ) ↔ ((x = A y = B z = 𝐶) ψ))
2017, 19syl6bb 185 . . . . . . 7 (((A V B V 𝐶 V) w = ⟨⟨A, B⟩, 𝐶⟩) → ((w = ⟨⟨x, y⟩, z φ) ↔ ((x = A y = B z = 𝐶) ψ)))
21203exbidv 1731 . . . . . 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)))
22 df-oprab 5440 . . . . . . . . . 10 {⟨⟨x, y⟩, z⟩ ∣ φ} = {wxyz(w = ⟨⟨x, y⟩, z φ)}
2322eleq2i 2086 . . . . . . . . 9 (w {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ w {wxyz(w = ⟨⟨x, y⟩, z φ)})
24 abid 2010 . . . . . . . . 9 (w {wxyz(w = ⟨⟨x, y⟩, z φ)} ↔ xyz(w = ⟨⟨x, y⟩, z φ))
2523, 24bitr2i 174 . . . . . . . 8 (xyz(w = ⟨⟨x, y⟩, z φ) ↔ w {⟨⟨x, y⟩, z⟩ ∣ φ})
26 eleq1 2082 . . . . . . . 8 (w = ⟨⟨A, B⟩, 𝐶⟩ → (w {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ}))
2725, 26syl5bb 181 . . . . . . 7 (w = ⟨⟨A, B⟩, 𝐶⟩ → (xyz(w = ⟨⟨x, y⟩, z φ) ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ}))
2827adantl 262 . . . . . 6 (((A V B V 𝐶 V) w = ⟨⟨A, B⟩, 𝐶⟩) → (xyz(w = ⟨⟨x, y⟩, z φ) ↔ ⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ}))
29 elisset 2545 . . . . . . . . . . 11 (A V → x x = A)
30 elisset 2545 . . . . . . . . . . 11 (B V → y y = B)
31 elisset 2545 . . . . . . . . . . 11 (𝐶 V → z z = 𝐶)
3229, 30, 313anim123i 1076 . . . . . . . . . 10 ((A V B V 𝐶 V) → (x x = A y y = B z z = 𝐶))
33 eeeanv 1790 . . . . . . . . . 10 (xyz(x = A y = B z = 𝐶) ↔ (x x = A y y = B z z = 𝐶))
3432, 33sylibr 137 . . . . . . . . 9 ((A V B V 𝐶 V) → xyz(x = A y = B z = 𝐶))
3534biantrurd 289 . . . . . . . 8 ((A V B V 𝐶 V) → (ψ ↔ (xyz(x = A y = B z = 𝐶) ψ)))
36 19.41vvv 1766 . . . . . . . 8 (xyz((x = A y = B z = 𝐶) ψ) ↔ (xyz(x = A y = B z = 𝐶) ψ))
3735, 36syl6rbbr 188 . . . . . . 7 ((A V B V 𝐶 V) → (xyz((x = A y = B z = 𝐶) ψ) ↔ ψ))
3837adantr 261 . . . . . 6 (((A V B V 𝐶 V) w = ⟨⟨A, B⟩, 𝐶⟩) → (xyz((x = A y = B z = 𝐶) ψ) ↔ ψ))
3921, 28, 383bitr3d 207 . . . . 5 (((A V B V 𝐶 V) w = ⟨⟨A, B⟩, 𝐶⟩) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ))
4039expcom 109 . . . 4 (w = ⟨⟨A, B⟩, 𝐶⟩ → ((A V B V 𝐶 V) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ)))
4140vtocleg 2601 . . 3 (⟨⟨A, B⟩, 𝐶 V → ((A V B V 𝐶 V) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ)))
427, 41mpcom 32 . 2 ((A V B V 𝐶 V) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ))
431, 2, 3, 42syl3an 1163 1 ((A 𝑉 B 𝑊 𝐶 𝑋) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 873   = wceq 1228  wex 1362   wcel 1374  {cab 2008  Vcvv 2535  cop 3353  {coprab 5437
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-oprab 5440
This theorem is referenced by:  eloprabg  5515  ovigg  5544
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