MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtoclegft Structured version   Visualization version   GIF version

Theorem vtoclegft 3311
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3312.) (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 3246 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 exim 1801 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑))
31, 2mpan9 485 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
433adant2 1100 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
5 19.9t 2109 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
653ad2ant2 1103 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → (∃𝑥𝜑𝜑))
74, 6mpbid 222 1 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054  wal 1521   = wceq 1523  wex 1744  wnf 1748  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233
This theorem is referenced by:  vtoclefex  33311
  Copyright terms: Public domain W3C validator