Theorem abssdv 3266

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 1896	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
3		abss 3261	. 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1370   ∈ wcel 2175  {cab 2190   ⊆ wss 3165

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-in 3171  df-ss 3178

This theorem is referenced by:  fmpt  5729  tfrlemibacc  6411  tfrlemibfn  6413  tfr1onlembacc  6427  tfr1onlembfn  6429  tfrcllembacc  6440  tfrcllembfn  6442  eroprf  6714  genipv  7621  hashfacen  10979  4sqlemafi  12660  4sqlemffi  12661  4sqleminfi  12662  4sqlem11  12666  lss1d  14087  lspsn  14120  metrest  14920