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Theorem bj-abvALT 34829
Description: Alternate version of bj-abv 34828; shorter but uses ax-8 2112. (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 1918 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 vexwt 2719 . . 3 (∀𝑥𝜑𝑦 ∈ {𝑥𝜑})
31, 2alrimih 1831 . 2 (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥𝜑})
4 eqv 3417 . 2 ({𝑥𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥𝜑})
53, 4sylibr 237 1 (∀𝑥𝜑 → {𝑥𝜑} = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1541   = wceq 1543  wcel 2110  {cab 2714  Vcvv 3408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410
This theorem is referenced by: (None)
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