Theorem raleq 2807

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

Assertion

Ref	Expression
raleq	|- (A = B -> (A.x e. A ph <-> A.x e. B ph))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   ph(x)

Proof of Theorem raleq

Step	Hyp	Ref	Expression
1		nfcv 2489	. 2 |- F/_xA
2		nfcv 2489	. 2 |- F/_xB
3	1, 2	raleqf 2803	1 |- (A = B -> (A.x e. A ph <-> A.x e. B ph))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  A.wral 2614

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 2478  df-ral 2619

This theorem is referenced by:  raleqi  2811  raleqdv  2813  raleqbi1dv  2815  sbralie  2848  r19.2zb  3640  inteq  3929  iineq1  3983  ncfinraise  4481  nnpweq  4523  fncnv  5158  isoeq4  5485  trd  5921  frd  5922  extd  5923  symd  5924  trrd  5925  refrd  5926  refd  5927  antird  5928  antid  5929  connexrd  5930  connexd  5931  iserd  5942  spacind  6287