Theorem currysetlem1 37307

Description: Lemma for currysetALT 37310. (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 2749	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = 𝑋
3	2	eleq2i 2832	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ 𝑋 ∈ 𝑋)
4		nfab1 2904	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}
5	1, 4	nfcxfr 2900	. . 3 ⊢ Ⅎ𝑥𝑋
6	5, 5	nfel 2916	. . . 4 ⊢ Ⅎ𝑥 𝑋 ∈ 𝑋
7		nfv 1921	. . . 4 ⊢ Ⅎ𝑥𝜑
8	6, 7	nfim 1903	. . 3 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑋 → 𝜑)
9		id 22	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10	9, 9	eleq12d 2834	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑥 ↔ 𝑋 ∈ 𝑋))
11	10	imbi1d 342	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑥 → 𝜑) ↔ (𝑋 ∈ 𝑋 → 𝜑)))
12	5, 8, 11	elabgf 3619	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ (𝑋 ∈ 𝑋 → 𝜑)))
13	3, 12	bitr3id 286	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  {cab 2718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-v 3434

This theorem is referenced by:  currysetlem2  37308  currysetlem3  37309