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Theorem bj-issetwt 33284
Description: Closed form of bj-issetw 33285. (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 df-clel 2761 . . 3 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}))
21a1i 11 . 2 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑})))
3 bj-vexwvt 33280 . . . . 5 (∀𝑥𝜑𝑧 ∈ {𝑥𝜑})
43biantrud 527 . . . 4 (∀𝑥𝜑 → (𝑧 = 𝐴 ↔ (𝑧 = 𝐴𝑧 ∈ {𝑥𝜑})))
54bicomd 214 . . 3 (∀𝑥𝜑 → ((𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}) ↔ 𝑧 = 𝐴))
65exbidv 2016 . 2 (∀𝑥𝜑 → (∃𝑧(𝑧 = 𝐴𝑧 ∈ {𝑥𝜑}) ↔ ∃𝑧 𝑧 = 𝐴))
7 bj-denotes 33282 . . 3 (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
87a1i 11 . 2 (∀𝑥𝜑 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
92, 6, 83bitrd 296 1 (∀𝑥𝜑 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦 𝑦 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  {cab 2751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-sb 2063  df-clab 2752  df-clel 2761
This theorem is referenced by:  bj-issetw  33285
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