Theorem currysetlem 37613

Description: Lemma for currysetlem 37613, 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 2930	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
2	1, 1	nfel 2942	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓}
3		nfv 1947	. . 3 ⊢ Ⅎ𝑥𝜑
4	2, 3	nfim 1929	. 2 ⊢ Ⅎ𝑥({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)
5		id 23	. . . 4 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → 𝑥 = {𝑥 ∣ 𝜓})
6	5, 5	eleq12d 2860	. . 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 3636	1 ⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 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:  curryset  37614