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Theorem vtoclegft 2619
 Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2620.) (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 2562 . . . 4 (A Bx x = A)
2 exim 1487 . . . 4 (x(x = Aφ) → (x x = Axφ))
31, 2mpan9 265 . . 3 ((A B x(x = Aφ)) → xφ)
433adant2 922 . 2 ((A B xφ x(x = Aφ)) → xφ)
5 19.9t 1530 . . 3 (Ⅎxφ → (xφφ))
653ad2ant2 925 . 2 ((A B xφ x(x = Aφ)) → (xφφ))
74, 6mpbid 135 1 ((A B xφ x(x = Aφ)) → φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 884  ∀wal 1240   = wceq 1242  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390 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-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553 This theorem is referenced by: (None)
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