Theorem abssdv 3314

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 1923	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
3		abss 3309	. 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1396   ∈ wcel 2205  {cab 2220   ⊆ wss 3213

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3219  df-ss 3226

This theorem is referenced by:  opabssxpd  4788  fmpt  5829  tfrlemibacc  6559  tfrlemibfn  6561  tfr1onlembacc  6575  tfr1onlembfn  6577  tfrcllembacc  6588  tfrcllembfn  6590  eroprf  6864  genipv  7826  hashfacen  11212  4sqlemafi  13097  4sqlemffi  13098  4sqleminfi  13099  4sqlem11  13103  lss1d  14548  lspsn  14581  metrest  15388