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Theorem bj-abvALT 36886
Description: Alternate version of bj-abv 36885; shorter but uses ax-8 2110. (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 1910 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 vexwt 2718 . . 3 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
31, 2alrimih 1824 . 2 (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 eqv 3489 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
53, 4sylibr 234 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wcel 2108  {cab 2713  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481
This theorem is referenced by: (None)
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