Theorem rabss3d 4035

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 1944	. 2 ⊢ Ⅎ𝑥𝜑
2		nfrab1 3436	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓}
3		nfrab1 3436	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐵 ∣ 𝜓}
4		rabss3d.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵)
5		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜓)
6	4, 5	jca 520	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 ∈ 𝐵 ∧ 𝜓))
7	6	ex 417	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜓)))
8		rabid 3437	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
9		rabid 3437	. . 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 3942	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143  {crab 3416   ⊆ wss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-ss 3922

This theorem is referenced by:  chnflenfi  18679  xpinpreima2  34297  reprss  35004  reprinfz1  35009  unitscyglem2  42963  unitscyglem4  42965  unitscyglem5  42966