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Theorem currysetlem1 34533
 Description: Lemma for currysetALT 34536. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
Hypothesis
Ref Expression
currysetlem2.def 𝑋 = {𝑥 ∣ (𝑥𝑥𝜑)}
Assertion
Ref Expression
currysetlem1 (𝑋𝑉 → (𝑋𝑋 ↔ (𝑋𝑋𝜑)))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝑉(𝑥)   𝑋(𝑥)

Proof of Theorem currysetlem1
StepHypRef Expression
1 currysetlem2.def . . . 4 𝑋 = {𝑥 ∣ (𝑥𝑥𝜑)}
21eqcomi 2807 . . 3 {𝑥 ∣ (𝑥𝑥𝜑)} = 𝑋
32eleq2i 2881 . 2 (𝑋 ∈ {𝑥 ∣ (𝑥𝑥𝜑)} ↔ 𝑋𝑋)
4 nfab1 2957 . . . 4 𝑥{𝑥 ∣ (𝑥𝑥𝜑)}
51, 4nfcxfr 2953 . . 3 𝑥𝑋
65, 5nfel 2969 . . . 4 𝑥 𝑋𝑋
7 nfv 1915 . . . 4 𝑥𝜑
86, 7nfim 1897 . . 3 𝑥(𝑋𝑋𝜑)
9 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
109, 9eleq12d 2884 . . . 4 (𝑥 = 𝑋 → (𝑥𝑥𝑋𝑋))
1110imbi1d 345 . . 3 (𝑥 = 𝑋 → ((𝑥𝑥𝜑) ↔ (𝑋𝑋𝜑)))
125, 8, 11elabgf 3611 . 2 (𝑋𝑉 → (𝑋 ∈ {𝑥 ∣ (𝑥𝑥𝜑)} ↔ (𝑋𝑋𝜑)))
133, 12bitr3id 288 1 (𝑋𝑉 → (𝑋𝑋 ↔ (𝑋𝑋𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3444 This theorem is referenced by:  currysetlem2  34534  currysetlem3  34535
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