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Theorem vtoclf 2908
 Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1986. (Contributed by NM, 30-Aug-1993.)
Hypotheses
Ref Expression
vtoclf.1 xψ
vtoclf.2 A V
vtoclf.3 (x = A → (φψ))
vtoclf.4 φ
Assertion
Ref Expression
vtoclf ψ
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem vtoclf
StepHypRef Expression
1 vtoclf.1 . . 3 xψ
2 vtoclf.2 . . . . 5 A V
32isseti 2865 . . . 4 x x = A
4 vtoclf.3 . . . . . 6 (x = A → (φψ))
54biimpd 198 . . . . 5 (x = A → (φψ))
65eximi 1576 . . . 4 (x x = Ax(φψ))
73, 6ax-mp 8 . . 3 x(φψ)
81, 719.36i 1872 . 2 (xφψ)
9 vtoclf.4 . 2 φ
108, 9mpg 1548 1 ψ
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  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:  vtocl  2909
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