Theorem ssrabdf 44484

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 3251	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
5		ssrabdf.2	. . 3 ⊢ Ⅎ𝑥𝐵
6		ssrabdf.1	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	ssrabf 44483	. 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓))
8	1, 4, 7	sylanbrc 581	1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 394  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2878  ∀wral 3057  {crab 3428   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rab 3429  df-v 3473  df-in 3954  df-ss 3964

This theorem is referenced by:  smfpimne2  46230