ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtoclegft GIF version

Theorem vtoclegft 2625
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2626.) (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 2568 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 exim 1490 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑))
31, 2mpan9 265 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
433adant2 923 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
5 19.9t 1533 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
653ad2ant2 926 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → (∃𝑥𝜑𝜑))
74, 6mpbid 135 1 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  w3a 885  wal 1241   = wceq 1243  wnf 1349  wex 1381  wcel 1393
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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator