Theorem rabss3d 30270

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 1911	. 2 ⊢ Ⅎ𝑥𝜑
2		nfrab1 3384	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓}
3		nfrab1 3384	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐵 ∣ 𝜓}
4		rabss3d.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵)
5		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜓)
6	4, 5	jca 514	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 ∈ 𝐵 ∧ 𝜓))
7	6	ex 415	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜓)))
8		rabid 3378	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
9		rabid 3378	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
10	7, 8, 9	3imtr4g 298	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}))
11	1, 2, 3, 10	ssrd 3971	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  {crab 3142   ⊆ wss 3935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-in 3942  df-ss 3951

This theorem is referenced by:  xpinpreima2  31145  reprss  31883  reprinfz1  31888