Theorem abexd 5288

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	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)
3	2	ex 412	. . . 4 ⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴))
4	3	alrimiv 1927	. . 3 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
5		abss 4034	. . 3 ⊢ ({𝑥 ∣ 𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
6	4, 5	sylibr 234	. 2 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)
7	1, 6	ssexd 5287	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  {cab 2708  Vcvv 3455   ⊆ wss 3922

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-rab 3412  df-v 3457  df-in 3929  df-ss 3939

This theorem is referenced by:  upfval2  49085