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Theorem vtoclegft 2926
 Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2927.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
Assertion
Ref Expression
vtoclegft ((A B xφ x(x = Aφ)) → φ)
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem vtoclegft
StepHypRef Expression
1 elisset 2869 . . . 4 (A Bx x = A)
2 exim 1575 . . . 4 (x(x = Aφ) → (x x = Axφ))
31, 2mpan9 455 . . 3 ((A B x(x = Aφ)) → xφ)
433adant2 974 . 2 ((A B xφ x(x = Aφ)) → xφ)
5 19.9t 1779 . . 3 (Ⅎxφ → (xφφ))
653ad2ant2 977 . 2 ((A B xφ x(x = Aφ)) → (xφφ))
74, 6mpbid 201 1 ((A B xφ x(x = Aφ)) → φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = 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-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  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: (None)
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