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Theorem vtocl2 2586
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
vtocl2.1 A V
vtocl2.2 B V
vtocl2.3 ((x = A y = B) → (φψ))
vtocl2.4 φ
Assertion
Ref Expression
vtocl2 ψ
Distinct variable groups:   x,y,A   x,B,y   ψ,x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem vtocl2
StepHypRef Expression
1 vtocl2.1 . . . . . 6 A V
21isseti 2541 . . . . 5 x x = A
3 vtocl2.2 . . . . . 6 B V
43isseti 2541 . . . . 5 y y = B
5 eeanv 1789 . . . . . 6 (xy(x = A y = B) ↔ (x x = A y y = B))
6 vtocl2.3 . . . . . . . 8 ((x = A y = B) → (φψ))
76biimpd 132 . . . . . . 7 ((x = A y = B) → (φψ))
872eximi 1474 . . . . . 6 (xy(x = A y = B) → xy(φψ))
95, 8sylbir 125 . . . . 5 ((x x = A y y = B) → xy(φψ))
102, 4, 9mp2an 404 . . . 4 xy(φψ)
11 nfv 1402 . . . . 5 yψ
121119.36-1 1545 . . . 4 (y(φψ) → (yφψ))
1310, 12eximii 1475 . . 3 x(yφψ)
141319.36aiv 1763 . 2 (xyφψ)
15 vtocl2.4 . . 3 φ
1615ax-gen 1318 . 2 yφ
1714, 16mpg 1320 1 ψ
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537
This theorem is referenced by:  caovord  5595
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