Theorem abexd 5253

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 413	. . 3 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴))
4	3	abssdv 3998	. 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)
5	1, 4	ssexd 5252	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  {cab 2717  Vcvv 3431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-in 3890  df-ss 3900

This theorem is referenced by:  upfval2  49667