Theorem rabssdv 3182

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)

Hypothesis

Ref	Expression
rabssdv.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
rabssdv	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵)

Distinct variable groups:   𝑥,𝐵   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rabssdv

Step	Hyp	Ref	Expression
1		rabssdv.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵)
2	1	3exp 1181	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝑥 ∈ 𝐵)))
3	2	ralrimiv 2507	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
4		rabss 3179	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
5	3, 4	sylibr 133	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 963   ∈ wcel 1481  ∀wral 2417  {crab 2421   ⊆ 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-3an 965  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-in 3082  df-ss 3089

This theorem is referenced by: (None)