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Theorem bj-issetwt 36876
Description: Closed form of bj-issetw 36877. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-issetwt (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
Distinct variable group:   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem bj-issetwt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfclel 2817 . . 3 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}))
21a1i 11 . 2 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑})))
3 vexwt 2719 . . . . 5 (∀𝑥𝜑𝑧 ∈ {𝑥𝜑})
43biantrud 531 . . . 4 (∀𝑥𝜑 → (𝑧 = 𝐴 ↔ (𝑧 = 𝐴𝑧 ∈ {𝑥𝜑})))
54bicomd 223 . . 3 (∀𝑥𝜑 → ((𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}) ↔ 𝑧 = 𝐴))
65exbidv 1921 . 2 (∀𝑥𝜑 → (∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}) ↔ ∃𝑧 𝑧 = 𝐴))
7 iseqsetv-clel 2820 . . 3 (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
87a1i 11 . 2 (∀𝑥𝜑 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
92, 6, 83bitrd 305 1 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-clel 2816
This theorem is referenced by:  bj-issetw  36877
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