Theorem rabss2 2132

Description: Subclass law for restricted abstraction.

Assertion

Ref	Expression
rabss2	⊢ (A ⊆ B → {x ∈ A∣φ} ⊆ {x ∈ B∣φ})

Distinct variable groups:   x,A   x,B

Proof of Theorem rabss2

Step	Hyp	Ref	Expression
1		pm3.45 564	. . . 4 ⊢ ((x ∈ A → x ∈ B) → ((x ∈ A ⋀ φ) → (x ∈ B ⋀ φ)))
2	1	19.20i 994	. . 3 ⊢ (∀x(x ∈ A → x ∈ B) → ∀x((x ∈ A ⋀ φ) → (x ∈ B ⋀ φ)))
3		ss2ab 2119	. . 3 ⊢ ({x∣(x ∈ A ⋀ φ)} ⊆ {x∣(x ∈ B ⋀ φ)} ↔ ∀x((x ∈ A ⋀ φ) → (x ∈ B ⋀ φ)))
4	2, 3	sylibr 200	. 2 ⊢ (∀x(x ∈ A → x ∈ B) → {x∣(x ∈ A ⋀ φ)} ⊆ {x∣(x ∈ B ⋀ φ)})
5		dfss2 2061	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
6		df-rab 1655	. . 3 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ⋀ φ)}
7		df-rab 1655	. . 3 ⊢ {x ∈ B∣φ} = {x∣(x ∈ B ⋀ φ)}
8	6, 7	sseq12i 2090	. 2 ⊢ ({x ∈ A∣φ} ⊆ {x ∈ B∣φ} ↔ {x∣(x ∈ A ⋀ φ)} ⊆ {x∣(x ∈ B ⋀ φ)})
9	4, 5, 8	3imtr4 219	1 ⊢ (A ⊆ B → {x ∈ A∣φ} ⊆ {x ∈ B∣φ})

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 956   ∈ wcel 960  {cab 1466  {crab 1651   ⊆ wss 2050

This theorem is referenced by:  shatomistic 10283

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655  df-in 2054  df-ss 2056

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