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

Theorem abvALT 3453
Description: Alternate proof of abv 3452, shorter but using more axioms. (Contributed by BJ, 19-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
abvALT ({𝑥𝜑} = V ↔ ∀𝑥𝜑)

Proof of Theorem abvALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-clab 2714 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
21albii 1820 . 2 (∀𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3 eqv 3450 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 sb8v 2348 . 2 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
52, 3, 43bitr4i 302 1 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1538   = wceq 1540  [wsb 2066  wcel 2105  {cab 2713  Vcvv 3441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator