Theorem ralss 3332

Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
ralss	⊢ (A ⊆ B → (∀x ∈ A φ ↔ ∀x ∈ B (x ∈ A → φ)))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem ralss

Step	Hyp	Ref	Expression
1		ssel 3267	. . . . 5 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	pm4.71rd 616	. . . 4 ⊢ (A ⊆ B → (x ∈ A ↔ (x ∈ B ∧ x ∈ A)))
3	2	imbi1d 308	. . 3 ⊢ (A ⊆ B → ((x ∈ A → φ) ↔ ((x ∈ B ∧ x ∈ A) → φ)))
4		impexp 433	. . 3 ⊢ (((x ∈ B ∧ x ∈ A) → φ) ↔ (x ∈ B → (x ∈ A → φ)))
5	3, 4	syl6bb 252	. 2 ⊢ (A ⊆ B → ((x ∈ A → φ) ↔ (x ∈ B → (x ∈ A → φ))))
6	5	ralbidv2 2636	1 ⊢ (A ⊆ B → (∀x ∈ A φ ↔ ∀x ∈ B (x ∈ A → φ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by: (None)