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Theorem currysetlem 36130
Description: Lemma for currysetlem 36130, where it is used with (𝑥𝑥𝜑) substituted for 𝜓. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
Assertion
Ref Expression
currysetlem ({𝑥𝜓} ∈ 𝑉 → ({𝑥𝜓} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} ↔ ({𝑥𝜓} ∈ {𝑥𝜓} → 𝜑)))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem currysetlem
StepHypRef Expression
1 nfab1 2904 . 2 𝑥{𝑥𝜓}
21, 1nfel 2916 . . 3 𝑥{𝑥𝜓} ∈ {𝑥𝜓}
3 nfv 1916 . . 3 𝑥𝜑
42, 3nfim 1898 . 2 𝑥({𝑥𝜓} ∈ {𝑥𝜓} → 𝜑)
5 id 22 . . . 4 (𝑥 = {𝑥𝜓} → 𝑥 = {𝑥𝜓})
65, 5eleq12d 2826 . . 3 (𝑥 = {𝑥𝜓} → (𝑥𝑥 ↔ {𝑥𝜓} ∈ {𝑥𝜓}))
76imbi1d 341 . 2 (𝑥 = {𝑥𝜓} → ((𝑥𝑥𝜑) ↔ ({𝑥𝜓} ∈ {𝑥𝜓} → 𝜑)))
81, 4, 7elabgf 3664 1 ({𝑥𝜓} ∈ 𝑉 → ({𝑥𝜓} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} ↔ ({𝑥𝜓} ∈ {𝑥𝜓} → 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  {cab 2708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475
This theorem is referenced by:  curryset  36131
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