Theorem rabss3d 4029

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 1947	. 2 ⊢ Ⅎ𝑥𝜑
2		nfrab1 3432	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓}
3		nfrab1 3432	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐵 ∣ 𝜓}
4		rabss3d.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵)
5		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜓)
6	4, 5	jca 521	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 ∈ 𝐵 ∧ 𝜓))
7	6	ex 418	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜓)))
8		rabid 3433	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
9		rabid 3433	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
10	7, 8, 9	3imtr4g 299	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}))
11	1, 2, 3, 10	ssrd 3936	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  {crab 3413   ⊆ wss 3899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-ss 3916

This theorem is used by:  chnflenfi  18795  xpinpreima2  34532  reprss  35239  reprinfz1  35244  unitscyglem2  43226  unitscyglem4  43228  unitscyglem5  43229