Theorem rabssdv 3259

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 1204	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝑥 ∈ 𝐵)))
3	2	ralrimiv 2566	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
4		rabss 3256	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
5	3, 4	sylibr 134	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 980   ∈ wcel 2164  ∀wral 2472  {crab 2476   ⊆ wss 3153

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-in 3159  df-ss 3166

This theorem is referenced by:  zsupssdc  12091