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Theorem bj-abeq2 33700
 Description: Remove dependency on ax-13 2344 from abeq2 2914. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-abeq2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-abeq2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax-5 1888 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
2 bj-hbab1 33698 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
31, 2cleqh 2906 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}))
4 abid 2779 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
54bibi2i 339 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
65albii 1801 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥(𝑥𝐴𝜑))
73, 6bitri 276 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207  ∀wal 1520   = wceq 1522   ∈ wcel 2081  {cab 2775 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863 This theorem is referenced by:  bj-abeq1  33701  bj-abbi2i  33703  bj-abbi2dv  33706  bj-clabel  33709  bj-ru1  33835
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