Theorem currysetlem1 36319

Description: Lemma for currysetALT 36322. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)

Hypothesis

Ref	Expression
currysetlem2.def	⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}

Assertion

Ref	Expression
currysetlem1	⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝑉(𝑥)   𝑋(𝑥)

Proof of Theorem currysetlem1

Step	Hyp	Ref	Expression
1		currysetlem2.def	. . . 4 ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}
2	1	eqcomi 2733	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = 𝑋
3	2	eleq2i 2817	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ 𝑋 ∈ 𝑋)
4		nfab1 2897	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}
5	1, 4	nfcxfr 2893	. . 3 ⊢ Ⅎ𝑥𝑋
6	5, 5	nfel 2909	. . . 4 ⊢ Ⅎ𝑥 𝑋 ∈ 𝑋
7		nfv 1909	. . . 4 ⊢ Ⅎ𝑥𝜑
8	6, 7	nfim 1891	. . 3 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑋 → 𝜑)
9		id 22	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10	9, 9	eleq12d 2819	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑥 ↔ 𝑋 ∈ 𝑋))
11	10	imbi1d 341	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑥 → 𝜑) ↔ (𝑋 ∈ 𝑋 → 𝜑)))
12	5, 8, 11	elabgf 3657	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ (𝑋 ∈ 𝑋 → 𝜑)))
13	3, 12	bitr3id 285	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cab 2701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-v 3468

This theorem is referenced by:  currysetlem2  36320  currysetlem3  36321