Theorem ssrabdf 45813

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 3264	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
5		ssrabdf.2	. . 3 ⊢ Ⅎ𝑥𝐵
6		ssrabdf.1	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	ssrabf 45812	. 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓))
8	1, 4, 7	sylanbrc 594	1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1813   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079  {crab 3416   ⊆ wss 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rab 3417  df-ss 3923

This theorem is referenced by:  smfpimne2  47534