Theorem raleq 2808

Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)

Assertion

Ref	Expression
raleq	⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem raleq

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxA
2		nfcv 2490	. 2 ⊢ ℲxB
3	1, 2	raleqf 2804	1 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∀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-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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  raleqi  2812  raleqdv  2814  raleqbi1dv  2816  sbralie  2849  r19.2zb  3641  inteq  3930  iineq1  3984  ncfinraise  4482  nnpweq  4524  fncnv  5159  isoeq4  5486  trd  5922  frd  5923  extd  5924  symd  5925  trrd  5926  refrd  5927  refd  5928  antird  5929  antid  5930  connexrd  5931  connexd  5932  iserd  5943  spacind  6288