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Theorem vtocl3 2604
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
vtocl3.1 A V
vtocl3.2 B V
vtocl3.3 𝐶 V
vtocl3.4 ((x = A y = B z = 𝐶) → (φψ))
vtocl3.5 φ
Assertion
Ref Expression
vtocl3 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)

Proof of Theorem vtocl3
StepHypRef Expression
1 vtocl3.1 . . . . . . 7 A V
21isseti 2557 . . . . . 6 x x = A
3 vtocl3.2 . . . . . . 7 B V
43isseti 2557 . . . . . 6 y y = B
5 vtocl3.3 . . . . . . 7 𝐶 V
65isseti 2557 . . . . . 6 z z = 𝐶
7 eeeanv 1805 . . . . . . 7 (xyz(x = A y = B z = 𝐶) ↔ (x x = A y y = B z z = 𝐶))
8 vtocl3.4 . . . . . . . . . 10 ((x = A y = B z = 𝐶) → (φψ))
98biimpd 132 . . . . . . . . 9 ((x = A y = B z = 𝐶) → (φψ))
109eximi 1488 . . . . . . . 8 (z(x = A y = B z = 𝐶) → z(φψ))
11102eximi 1489 . . . . . . 7 (xyz(x = A y = B z = 𝐶) → xyz(φψ))
127, 11sylbir 125 . . . . . 6 ((x x = A y y = B z z = 𝐶) → xyz(φψ))
132, 4, 6, 12mp3an 1231 . . . . 5 xyz(φψ)
14 nfv 1418 . . . . . . 7 zψ
151419.36-1 1560 . . . . . 6 (z(φψ) → (zφψ))
16152eximi 1489 . . . . 5 (xyz(φψ) → xy(zφψ))
1713, 16ax-mp 7 . . . 4 xy(zφψ)
18 nfv 1418 . . . . 5 yψ
191819.36-1 1560 . . . 4 (y(zφψ) → (yzφψ))
2017, 19eximii 1490 . . 3 x(yzφψ)
212019.36aiv 1778 . 2 (xyzφψ)
22 vtocl3.5 . . 3 φ
2322gen2 1336 . 2 yzφ
2421, 23mpg 1337 1 ψ
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 884  wal 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551
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: (None)
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