Theorem currysetlem2 36914

Description: Lemma for currysetALT 36916. (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 36913	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))
3	2	biimpd 229	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → (𝑋 ∈ 𝑋 → 𝜑)))
4	3	pm2.43d 53	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {cab 2717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490

This theorem is referenced by:  currysetlem3  36915