Theorem ssrabdv 3327

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)

Hypotheses

Ref	Expression
ssrabdv.1	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
ssrabdv.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem ssrabdv

Step	Hyp	Ref	Expression
1		ssrabdv.1	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
2		ssrabdv.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)
3	2	ralrimiva 2623	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
4		ssrab 3326	. 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 3, 4	sylanbrc 421	1 ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2209  ∀wral 2528  {crab 2532   ⊆ wss 3220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-in 3226  df-ss 3233

This theorem is referenced by:  perfectlem2  16097