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Theorem bj-abvALT 36385
Description: Alternate version of bj-abv 36384; shorter but uses ax-8 2101. (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 2710 . . 3 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
31, 2alrimih 1819 . 2 (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 eqv 3480 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
53, 4sylibr 233 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1532   = wceq 1534  wcel 2099  {cab 2705  Vcvv 3471
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473
This theorem is referenced by: (None)
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