Theorem rabss2 2125

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		pm3.45 561	. . . 4 ⊢ ((x ∈ A → x ∈ B) → ((x ∈ A ⋀ φ) → (x ∈ B ⋀ φ)))
2	1	19.20i 990	. . 3 ⊢ (∀x(x ∈ A → x ∈ B) → ∀x((x ∈ A ⋀ φ) → (x ∈ B ⋀ φ)))
3		ss2ab 2112	. . 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 2054	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
6		df-rab 1649	. . 3 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ⋀ φ)}
7		df-rab 1649	. . 3 ⊢ {x ∈ B∣φ} = {x∣(x ∈ B ⋀ φ)}
8	6, 7	sseq12i 2083	. 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 952   ∈ wcel 956  {cab 1461  {crab 1645   ⊆ wss 2043

This theorem is referenced by:  shatomistic 10225

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-rab 1649  df-in 2047  df-ss 2049

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