Theorem rabssrabd 4106

Description: Subclass of a restricted class abstraction. (Contributed by AV, 4-Jun-2022.)

Hypotheses

Ref	Expression
rabssrabd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
rabssrabd.2	⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

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

Proof of Theorem rabssrabd

Step	Hyp	Ref	Expression
1		3anan32 1097	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓))
2		rabssrabd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒)
3	1, 2	sylbir 235	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
4	3	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
5	4	ss2rabdv 4099	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})
6		rabssrabd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7		rabss2 4101	. . 3 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒})
8	6, 7	syl 17	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒})
9	5, 8	sstrd 4019	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  {crab 3443   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rab 3444  df-ss 3993

This theorem is referenced by:  suppfnss  8230  clwlknon2num  30400  numclwlk1lem2  30402