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Theorem spc3gv 2639
Description: Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
Hypothesis
Ref Expression
spc3egv.1 ((x = A y = B z = 𝐶) → (φψ))
Assertion
Ref Expression
spc3gv ((A 𝑉 B 𝑊 𝐶 𝑋) → (xyzφψ))
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 spc3gv
StepHypRef Expression
1 elisset 2562 . . . 4 (A 𝑉x x = A)
2 elisset 2562 . . . 4 (B 𝑊y y = B)
3 elisset 2562 . . . 4 (𝐶 𝑋z z = 𝐶)
41, 2, 33anim123i 1088 . . 3 ((A 𝑉 B 𝑊 𝐶 𝑋) → (x x = A y y = B z z = 𝐶))
5 eeeanv 1805 . . 3 (xyz(x = A y = B z = 𝐶) ↔ (x x = A y y = B z z = 𝐶))
64, 5sylibr 137 . 2 ((A 𝑉 B 𝑊 𝐶 𝑋) → xyz(x = A y = B z = 𝐶))
7 spc3egv.1 . . . . . . . 8 ((x = A y = B z = 𝐶) → (φψ))
87biimpcd 148 . . . . . . 7 (φ → ((x = A y = B z = 𝐶) → ψ))
982alimi 1342 . . . . . 6 (yzφyz((x = A y = B z = 𝐶) → ψ))
109alimi 1341 . . . . 5 (xyzφxyz((x = A y = B z = 𝐶) → ψ))
11 exim 1487 . . . . . 6 (z((x = A y = B z = 𝐶) → ψ) → (z(x = A y = B z = 𝐶) → zψ))
12112alimi 1342 . . . . 5 (xyz((x = A y = B z = 𝐶) → ψ) → xy(z(x = A y = B z = 𝐶) → zψ))
1310, 12syl 14 . . . 4 (xyzφxy(z(x = A y = B z = 𝐶) → zψ))
14 exim 1487 . . . . 5 (y(z(x = A y = B z = 𝐶) → zψ) → (yz(x = A y = B z = 𝐶) → yzψ))
1514alimi 1341 . . . 4 (xy(z(x = A y = B z = 𝐶) → zψ) → x(yz(x = A y = B z = 𝐶) → yzψ))
16 exim 1487 . . . 4 (x(yz(x = A y = B z = 𝐶) → yzψ) → (xyz(x = A y = B z = 𝐶) → xyzψ))
1713, 15, 163syl 17 . . 3 (xyzφ → (xyz(x = A y = B z = 𝐶) → xyzψ))
18 19.9v 1748 . . . 4 (xyzψyzψ)
19 19.9v 1748 . . . 4 (yzψzψ)
20 19.9v 1748 . . . 4 (zψψ)
2118, 19, 203bitri 195 . . 3 (xyzψψ)
2217, 21syl6ib 150 . 2 (xyzφ → (xyz(x = A y = B z = 𝐶) → ψ))
236, 22syl5com 26 1 ((A 𝑉 B 𝑊 𝐶 𝑋) → (xyzφψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 884  wal 1240   = wceq 1242  wex 1378   wcel 1390
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  funopg  4877
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