Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-abvALT Structured version   Visualization version   GIF version

Theorem bj-abvALT 36850
Description: Alternate version of bj-abv 36849; shorter but uses ax-8 2106. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-abvALT (∀𝑥𝜑 → {𝑥𝜑} = V)

Proof of Theorem bj-abvALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax-5 1906 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 vexwt 2715 . . 3 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
31, 2alrimih 1819 . 2 (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 eqv 3487 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
53, 4sylibr 234 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1533   = wceq 1535  wcel 2104  {cab 2710  Vcvv 3477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-v 3479
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator