Theorem currysetlem2 35064

Description: Lemma for currysetALT 35066. (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 35063	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))
3	2	biimpd 228	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → (𝑋 ∈ 𝑋 → 𝜑)))
4	3	pm2.43d 53	1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {cab 2715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-v 3424

This theorem is referenced by:  currysetlem3  35065