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Theorem vtoclb 2605
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 224 . 2 (x = A → ((φψ) ↔ (χθ)))
5 vtoclb.4 . 2 (φψ)
61, 4, 5vtocl 2602 1 (χθ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  Vcvv 2551
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  alexeq  2664  sbss  3323
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