Theorem ralss 3000

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 2933	. . . . 5 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	pm4.71rd 374	. . . 4 ⊢ (A ⊆ B → (x ∈ A ↔ (x ∈ B ∧ x ∈ A)))
3	2	imbi1d 220	. . 3 ⊢ (A ⊆ B → ((x ∈ A → φ) ↔ ((x ∈ B ∧ x ∈ A) → φ)))
4		impexp 250	. . 3 ⊢ (((x ∈ B ∧ x ∈ A) → φ) ↔ (x ∈ B → (x ∈ A → φ)))
5	3, 4	syl6bb 185	. 2 ⊢ (A ⊆ B → ((x ∈ A → φ) ↔ (x ∈ B → (x ∈ A → φ))))
6	5	ralbidv2 2322	1 ⊢ (A ⊆ B → (∀x ∈ A φ ↔ ∀x ∈ B (x ∈ A → φ)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925

This theorem is referenced by: (None)