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Theorem bj-issetwt 36580
Description: Closed form of bj-issetw 36581. (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 2804 . . 3 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}))
21a1i 11 . 2 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑})))
3 vexwt 2708 . . . . 5 (∀𝑥𝜑𝑧 ∈ {𝑥𝜑})
43biantrud 530 . . . 4 (∀𝑥𝜑 → (𝑧 = 𝐴 ↔ (𝑧 = 𝐴𝑧 ∈ {𝑥𝜑})))
54bicomd 222 . . 3 (∀𝑥𝜑 → ((𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}) ↔ 𝑧 = 𝐴))
65exbidv 1917 . 2 (∀𝑥𝜑 → (∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}) ↔ ∃𝑧 𝑧 = 𝐴))
7 bj-denotes 36577 . . 3 (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
87a1i 11 . 2 (∀𝑥𝜑 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
92, 6, 83bitrd 304 1 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1532   = wceq 1534  wex 1774  wcel 2099  {cab 2703
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
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-clel 2803
This theorem is referenced by:  bj-issetw  36581
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