Theorem rabss3d 28736

Description: Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)

Hypothesis

Ref	Expression
rabss3d.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
rabss3d	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓})

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

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabss3d

Step	Hyp	Ref	Expression
1		nfv 1830	. 2 ⊢ Ⅎ𝑥𝜑
2		nfrab1 3099	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓}
3		nfrab1 3099	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐵 ∣ 𝜓}
4		rabss3d.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵)
5		simprr 792	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜓)
6	4, 5	jca 553	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 ∈ 𝐵 ∧ 𝜓))
7	6	ex 449	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜓)))
8		rabid 3095	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
9		rabid 3095	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
10	7, 8, 9	3imtr4g 284	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}))
11	1, 2, 3, 10	ssrd 3573	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  {crab 2900   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547  df-ss 3554

This theorem is referenced by:  xpinpreima2  29281