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Theorem vtoclgaf 2919
 Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtoclgaf.1 xA
vtoclgaf.2 xψ
vtoclgaf.3 (x = A → (φψ))
vtoclgaf.4 (x Bφ)
Assertion
Ref Expression
vtoclgaf (A Bψ)
Distinct variable group:   x,B
Allowed substitution hints:   φ(x)   ψ(x)   A(x)

Proof of Theorem vtoclgaf
StepHypRef Expression
1 vtoclgaf.1 . . 3 xA
21nfel1 2499 . . . 4 x A B
3 vtoclgaf.2 . . . 4 xψ
42, 3nfim 1813 . . 3 x(A Bψ)
5 eleq1 2413 . . . 4 (x = A → (x BA B))
6 vtoclgaf.3 . . . 4 (x = A → (φψ))
75, 6imbi12d 311 . . 3 (x = A → ((x Bφ) ↔ (A Bψ)))
8 vtoclgaf.4 . . 3 (x Bφ)
91, 4, 7, 8vtoclgf 2913 . 2 (A B → (A Bψ))
109pm2.43i 43 1 (A Bψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 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-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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:  vtoclga  2920  ssiun2s  4010  fvmptss  5705  fvmptf  5722
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