Theorem ssrabdv 3346

Description: Subclass of a restricted class abstraction (deduction rule). (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 2698	. 2 ⊢ (φ → ∀x ∈ B ψ)
4		ssrab 3345	. 2 ⊢ (B ⊆ {x ∈ A ∣ ψ} ↔ (B ⊆ A ∧ ∀x ∈ B ψ))
5	1, 3, 4	sylanbrc 645	1 ⊢ (φ → B ⊆ {x ∈ A ∣ ψ})

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2615  {crab 2619   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)