Theorem currysetlem 34278

Description: Lemma for currysetlem 34278, 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 2978	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
2	1, 1	nfel 2991	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓}
3		nfv 1914	. . 3 ⊢ Ⅎ𝑥𝜑
4	2, 3	nfim 1896	. 2 ⊢ Ⅎ𝑥({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)
5		id 22	. . . 4 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → 𝑥 = {𝑥 ∣ 𝜓})
6	5, 5	eleq12d 2906	. . 3 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → (𝑥 ∈ 𝑥 ↔ {𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓}))
7	6	imbi1d 344	. 2 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → ((𝑥 ∈ 𝑥 → 𝜑) ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)))
8	1, 4, 7	elabgf 3660	1 ⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ∈ wcel 2113  {cab 2798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-v 3493

This theorem is referenced by:  curryset  34279