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Theorem vtoclegft 3500
 Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3501.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
Assertion
Ref Expression
vtoclegft ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem vtoclegft
StepHypRef Expression
1 elisset 2833 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 exim 1835 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑))
31, 2mpan9 510 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
433adant2 1128 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
5 19.9t 2202 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
653ad2ant2 1131 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → (∃𝑥𝜑𝜑))
74, 6mpbid 235 1 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-clel 2830 This theorem is referenced by:  vtoclefex  35031
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