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Theorem vtocl2 2910
 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 2865 . . . . 5 x x = A
3 vtocl2.2 . . . . . 6 B V
43isseti 2865 . . . . 5 y y = B
5 eeanv 1913 . . . . . 6 (xy(x = A y = B) ↔ (x x = A y y = B))
6 vtocl2.3 . . . . . . . 8 ((x = A y = B) → (φψ))
76biimpd 198 . . . . . . 7 ((x = A y = B) → (φψ))
872eximi 1577 . . . . . 6 (xy(x = A y = B) → xy(φψ))
95, 8sylbir 204 . . . . 5 ((x x = A y y = B) → xy(φψ))
102, 4, 9mp2an 653 . . . 4 xy(φψ)
11 19.36v 1896 . . . . 5 (y(φψ) ↔ (yφψ))
1211exbii 1582 . . . 4 (xy(φψ) ↔ x(yφψ))
1310, 12mpbi 199 . . 3 x(yφψ)
141319.36aiv 1897 . 2 (xyφψ)
15 vtocl2.4 . . 3 φ
1615ax-gen 1546 . 2 yφ
1714, 16mpg 1548 1 ψ
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  caovord  5629
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