Theorem ssralv 2998

Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)

Assertion

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

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem ssralv

Step	Hyp	Ref	Expression
1		ssel 2933	. . 3 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	imim1d 69	. 2 ⊢ (A ⊆ B → ((x ∈ B → φ) → (x ∈ A → φ)))
3	2	ralimdv2 2383	1 ⊢ (A ⊆ B → (∀x ∈ B φ → ∀x ∈ A φ))

Syntax hints:   → wi 4   ∈ 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:  iinss1  3660  poss  4026  sess2  4060  trssord  4083  funco  4883  funimaexglem  4925  isores3  5398  isoini2  5401  smores  5848  smores2  5850  tfrlem5  5871  peano5nni  7698