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Theorem vtoclb 2912
 Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
Hypotheses
Ref Expression
vtoclb.1 A V
vtoclb.2 (x = A → (φχ))
vtoclb.3 (x = A → (ψθ))
vtoclb.4 (φψ)
Assertion
Ref Expression
vtoclb (χθ)
Distinct variable groups:   x,A   χ,x   θ,x
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem vtoclb
StepHypRef Expression
1 vtoclb.1 . 2 A V
2 vtoclb.2 . . 3 (x = A → (φχ))
3 vtoclb.3 . . 3 (x = A → (ψθ))
42, 3bibi12d 312 . 2 (x = A → ((φψ) ↔ (χθ)))
5 vtoclb.4 . 2 (φψ)
61, 4, 5vtocl 2909 1 (χθ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = 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-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:  alexeq  2968  sbss  3659
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