Theorem currysetlem2 34283

Description: Lemma for currysetALT 34285. (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
currysetlem2	⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑))

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem currysetlem2

Step	Hyp	Ref	Expression
1		currysetlem2.def	. . . 4 ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}
2	1	currysetlem1 34282	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))
3	2	biimpd 231	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → (𝑋 ∈ 𝑋 → 𝜑)))
4	3	pm2.43d 53	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑))

Syntax hints:   → wi 4   = 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:  currysetlem3  34284