Theorem rsp 2674

Description: Restricted specialization. (Contributed by NM, 17-Oct-1996.)

Assertion

Ref	Expression
rsp	⊢ (∀x ∈ A φ → (x ∈ A → φ))

Proof of Theorem rsp

Step	Hyp	Ref	Expression
1		df-ral 2619	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
2		sp 1747	. 2 ⊢ (∀x(x ∈ A → φ) → (x ∈ A → φ))
3	1, 2	sylbi 187	1 ⊢ (∀x ∈ A φ → (x ∈ A → φ))

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710  ∀wral 2614

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

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-ral 2619

This theorem is referenced by:  rsp2  2676  rspec  2678  r19.12  2727  ralbi  2750  reupick2  3541  dfiun2g  3999  iinss2  4018  pw1disj  4167  sfinltfin  4535  fun11iun  5305  chfnrn  5399  ffnfv  5427  mpteq12f  5655  mpt2eq123  5661  fvmptss  5705