Theorem abssdv 3341

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Hypothesis

Ref	Expression
abssdv.1	⊢ (φ → (ψ → x ∈ A))

Assertion

Ref	Expression
abssdv	⊢ (φ → {x ∣ ψ} ⊆ A)

Distinct variable groups:   φ,x   x,A

Allowed substitution hint:   ψ(x)

Proof of Theorem abssdv

Step	Hyp	Ref	Expression
1		abssdv.1	. . 3 ⊢ (φ → (ψ → x ∈ A))
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀x(ψ → x ∈ A))
3		abss 3336	. 2 ⊢ ({x ∣ ψ} ⊆ A ↔ ∀x(ψ → x ∈ A))
4	2, 3	sylibr 203	1 ⊢ (φ → {x ∣ ψ} ⊆ A)

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710  {cab 2339   ⊆ 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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  vfinspeqtncv  4554  fmpt  5693