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Theorem bj-abvALT 36294
Description: Alternate version of bj-abv 36293; shorter but uses ax-8 2100. (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 1905 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 vexwt 2708 . . 3 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
31, 2alrimih 1818 . 2 (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 eqv 3477 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
53, 4sylibr 233 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531   = wceq 1533  wcel 2098  {cab 2703  Vcvv 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470
This theorem is referenced by: (None)
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