Theorem ssrabdv 2129

Description: Subclass of a restricted class abstraction (deduction rule).

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

Proof of Theorem ssrabdv

Step	Hyp	Ref	Expression
1		ssrabdv.1	. . 3 ⊢ (φ → B ⊆ A)
2		ssrabdv.2	. . . 4 ⊢ ((φ ⋀ x ∈ B) → ψ)
3	2	r19.21aiva 1717	. . 3 ⊢ (φ → ∀x ∈ B ψ)
4	1, 3	jca 288	. 2 ⊢ (φ → (B ⊆ A ⋀ ∀x ∈ B ψ))
5		ssrab 2128	. 2 ⊢ (B ⊆ {x ∈ A∣ψ} ↔ (B ⊆ A ⋀ ∀x ∈ B ψ))
6	4, 5	sylibr 200	1 ⊢ (φ → B ⊆ {x ∈ A∣ψ})

Syntax hints:   → wi 3   ⋀ wa 223   ∈ wcel 960  ∀wral 1648  {crab 1651   ⊆ wss 2050

This theorem is referenced by:  blss 7850

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rab 1655  df-in 2054  df-ss 2056

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