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Theorem currysetlem 36947
Description: Lemma for currysetlem 36947, 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 2906 . 2 𝑥{𝑥𝜓}
21, 1nfel 2919 . . 3 𝑥{𝑥𝜓} ∈ {𝑥𝜓}
3 nfv 1913 . . 3 𝑥𝜑
42, 3nfim 1895 . 2 𝑥({𝑥𝜓} ∈ {𝑥𝜓} → 𝜑)
5 id 22 . . . 4 (𝑥 = {𝑥𝜓} → 𝑥 = {𝑥𝜓})
65, 5eleq12d 2834 . . 3 (𝑥 = {𝑥𝜓} → (𝑥𝑥 ↔ {𝑥𝜓} ∈ {𝑥𝜓}))
76imbi1d 341 . 2 (𝑥 = {𝑥𝜓} → ((𝑥𝑥𝜑) ↔ ({𝑥𝜓} ∈ {𝑥𝜓} → 𝜑)))
81, 4, 7elabgf 3673 1 ({𝑥𝜓} ∈ 𝑉 → ({𝑥𝜓} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} ↔ ({𝑥𝜓} ∈ {𝑥𝜓} → 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  {cab 2713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-v 3481
This theorem is referenced by:  curryset  36948
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