Theorem currysetlem 34757

Description: Lemma for currysetlem 34757, where it is used with (𝑥 ∈ 𝑥 → 𝜑) substituted for 𝜓. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)

Assertion

Ref	Expression
currysetlem	⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)))

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem currysetlem

Step	Hyp	Ref	Expression
1		nfab1 2901	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
2	1, 1	nfel 2913	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓}
3		nfv 1921	. . 3 ⊢ Ⅎ𝑥𝜑
4	2, 3	nfim 1903	. 2 ⊢ Ⅎ𝑥({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)
5		id 22	. . . 4 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → 𝑥 = {𝑥 ∣ 𝜓})
6	5, 5	eleq12d 2827	. . 3 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → (𝑥 ∈ 𝑥 ↔ {𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓}))
7	6	imbi1d 345	. 2 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → ((𝑥 ∈ 𝑥 → 𝜑) ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)))
8	1, 4, 7	elabgf 3568	1 ⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1542   ∈ wcel 2114  {cab 2716

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3400

This theorem is referenced by:  curryset  34758