Theorem ssrabdf 45085

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by Glauco Siliprandi, 5-Jan-2025.)

Hypotheses

Ref	Expression
ssrabdf.1	⊢ Ⅎ𝑥𝐴
ssrabdf.2	⊢ Ⅎ𝑥𝐵
ssrabdf.3	⊢ Ⅎ𝑥𝜑
ssrabdf.4	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
ssrabdf.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)

Assertion

Ref	Expression
ssrabdf	⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

Proof of Theorem ssrabdf

Step	Hyp	Ref	Expression
1		ssrabdf.4	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
2		ssrabdf.3	. . 3 ⊢ Ⅎ𝑥𝜑
3		ssrabdf.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)
4	2, 3	ralrimia 3258	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
5		ssrabdf.2	. . 3 ⊢ Ⅎ𝑥𝐵
6		ssrabdf.1	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	ssrabf 45084	. 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓))
8	1, 4, 7	sylanbrc 583	1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1782   ∈ wcel 2108  Ⅎwnfc 2890  ∀wral 3061  {crab 3436   ⊆ wss 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-ss 3983

This theorem is referenced by:  smfpimne2  46824