Theorem abssdv 3176

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 1847	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
3		abss 3171	. 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓 → 𝑥 ∈ 𝐴))
4	2, 3	sylibr 133	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1330   ∈ wcel 1481  {cab 2126   ⊆ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3082  df-ss 3089

This theorem is referenced by:  fmpt  5578  tfrlemibacc  6231  tfrlemibfn  6233  tfr1onlembacc  6247  tfr1onlembfn  6249  tfrcllembacc  6260  tfrcllembfn  6262  eroprf  6530  genipv  7341  hashfacen  10611  metrest  12714