Theorem currysetlem 37299

Description: Lemma for currysetlem 37299, 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 2904	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
2	1, 1	nfel 2916	. . 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 2834	. . 3 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → (𝑥 ∈ 𝑥 ↔ {𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓}))
7	6	imbi1d 342	. 2 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → ((𝑥 ∈ 𝑥 → 𝜑) ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)))
8	1, 4, 7	elabgf 3619	1 ⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 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:  curryset  37300