Theorem abexd 5290

Description: Conditions for a class abstraction to be a set, deduction form. (Contributed by AV, 19-Apr-2025.)

Hypotheses

Ref	Expression
abexd.1	⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
abexd.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
abexd	⊢ (𝜑 → {𝑥 ∣ 𝜓} ∈ V)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

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

Proof of Theorem abexd

Step	Hyp	Ref	Expression
1		abexd.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		abexd.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
3	2	ex 418	. . 3 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴))
4	3	abssdv 4015	. 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)
5	1, 4	ssexd 5289	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  {cab 2738  Vcvv 3450

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 3906  df-ss 3916

This theorem is used by:  upfval2  50104