Theorem currysetlem1 37624

Description: Lemma for currysetALT 37627. (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 2775	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = 𝑋
3	2	eleq2i 2858	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ 𝑋 ∈ 𝑋)
4		nfab1 2930	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}
5	1, 4	nfcxfr 2926	. . 3 ⊢ Ⅎ𝑥𝑋
6	5, 5	nfel 2942	. . . 4 ⊢ Ⅎ𝑥 𝑋 ∈ 𝑋
7		nfv 1947	. . . 4 ⊢ Ⅎ𝑥𝜑
8	6, 7	nfim 1929	. . 3 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑋 → 𝜑)
9		id 23	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10	9, 9	eleq12d 2860	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑥 ↔ 𝑋 ∈ 𝑋))
11	10	imbi1d 344	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑥 → 𝜑) ↔ (𝑋 ∈ 𝑋 → 𝜑)))
12	5, 8, 11	elabgf 3636	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ (𝑋 ∈ 𝑋 → 𝜑)))
13	3, 12	bitr3id 288	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  {cab 2744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-v 3460

This theorem is used by:  currysetlem2  37625  currysetlem3  37626