Theorem rsp 2675

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 2620	. 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 2615

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-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-ral 2620

This theorem is referenced by:  rsp2  2677  rspec  2679  r19.12  2728  ralbi  2751  reupick2  3542  dfiun2g  4000  iinss2  4019  pw1disj  4168  sfinltfin  4536  fun11iun  5306  chfnrn  5400  ffnfv  5428  mpteq12f  5656  mpt2eq123  5662  fvmptss  5706