Theorem rabssdv 3347

Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)

Hypothesis

Ref	Expression
rabssdv.1	⊢ ((φ ∧ x ∈ A ∧ ψ) → x ∈ B)

Assertion

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

Distinct variable groups:   x,B   φ,x

Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem rabssdv

Step	Hyp	Ref	Expression
1		rabssdv.1	. . . 4 ⊢ ((φ ∧ x ∈ A ∧ ψ) → x ∈ B)
2	1	3exp 1150	. . 3 ⊢ (φ → (x ∈ A → (ψ → x ∈ B)))
3	2	ralrimiv 2697	. 2 ⊢ (φ → ∀x ∈ A (ψ → x ∈ B))
4		rabss 3344	. 2 ⊢ ({x ∈ A ∣ ψ} ⊆ B ↔ ∀x ∈ A (ψ → x ∈ B))
5	3, 4	sylibr 203	1 ⊢ (φ → {x ∈ A ∣ ψ} ⊆ B)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 934   ∈ wcel 1710  ∀wral 2615  {crab 2619   ⊆ wss 3258

This proof depends on 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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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 used by: (None)