Theorem ss2rabd 3997

Description: Subclass of a restricted class abstraction (deduction form). Saves ax-10 2177, ax-11 2193, ax-12 2212 over using ss2rab 3994 and sylibr 237. (Contributed by SN, 4-Feb-2026.)

Hypothesis

Ref	Expression
ss2rabd.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))

Assertion

Ref	Expression
ss2rabd	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

Proof of Theorem ss2rabd

Step	Hyp	Ref	Expression
1		ss2rabd.1	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
2		df-ral 3077	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3		imdistan 578	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → (𝜓 → 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	3	albii 1851	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒)))
5	2, 4	bitri 278	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒)))
6	1, 5	sylib 221	. . 3 ⊢ (𝜑 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒)))
7		ss2abim 3985	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒)) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)})
8	6, 7	syl 18	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)})
9		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
10		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)}
11	8, 9, 10	3sstr4g 3961	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   ∈ wcel 2144  {cab 2738  ∀wral 3076  {crab 3412   ⊆ wss 3895

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-ral 3077  df-rab 3413  df-ss 3896

This theorem is used by:  ss2rabdv  4000  ondomon  10648  xrlimcnp  27297  ss2rabdf  46241  pimiooltgt  47796