Theorem rabssdv 3307

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 1228	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝑥 ∈ 𝐵)))
3	2	ralrimiv 2604	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 ∈ 𝐵))
4		rabss 3304	. 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 1004   ∈ wcel 2202  ∀wral 2510  {crab 2514   ⊆ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-in 3206  df-ss 3213

This theorem is referenced by:  zsupssdc  10497