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

Theorem vtoclegftOLD 3572
Description: Obsolete version of vtoclegft 3571 as of 26-Jan-2025. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
vtoclegftOLD ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem vtoclegftOLD
StepHypRef Expression
1 elisset 2815 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 exim 1836 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑))
31, 2mpan9 507 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
433adant2 1131 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
5 19.9t 2197 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
653ad2ant2 1134 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → (∃𝑥𝜑𝜑))
74, 6mpbid 231 1 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087  wal 1539   = wceq 1541  wex 1781  wnf 1785  wcel 2106
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-clel 2810
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator