Theorem abssdv 3322

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 1927	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
3		abss 3317	. 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4  ∀wal 1400   ∈ wcel 2209  {cab 2224   ⊆ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233

This theorem is used by:  opabssxpd  4813  fmpt  5861  tfrlemibacc  6601  tfrlemibfn  6603  tfr1onlembacc  6617  tfr1onlembfn  6619  tfrcllembacc  6630  tfrcllembfn  6632  eroprf  6906  genipv  7882  hashfacen  11306  hashf1lem2  11308  4sqlemafi  13203  4sqlemffi  13204  4sqleminfi  13205  4sqlem11  13209  lss1d  14829  lspsn  14862  metrest  15732