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Theorem vtocld 2907
 Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
vtocld.1 (φA V)
vtocld.2 ((φ x = A) → (ψχ))
vtocld.3 (φψ)
Assertion
Ref Expression
vtocld (φχ)
Distinct variable groups:   x,A   φ,x   χ,x
Allowed substitution hints:   ψ(x)   V(x)

Proof of Theorem vtocld
StepHypRef Expression
1 vtocld.1 . 2 (φA V)
2 vtocld.2 . 2 ((φ x = A) → (ψχ))
3 vtocld.3 . 2 (φψ)
4 nfv 1619 . 2 xφ
5 nfcvd 2490 . 2 (φxA)
6 nfvd 1620 . 2 (φ → Ⅎxχ)
71, 2, 3, 4, 5, 6vtocldf 2906 1 (φχ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710 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-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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