Theorem raleqi 2503

Description: Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
raleq1i.1	|- A = B

Assertion

Ref	Expression
raleqi	|- (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 raleqi

Step	Hyp	Ref	Expression
1		raleq1i.1	. 2 |- A = B
2		raleq 2499	. 2 |- (A = B -> (A.x e. A ph <-> A.x e. B ph))
3	1, 2	ax-mp 7	1 |- (A.x e. A ph <-> A.x e. B ph)

Syntax hints:   <-> wb 98   = wceq 1242  A.wral 2300

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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  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-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305

This theorem is referenced by:  ralrab2  2700  ralprg  3412  raltpg  3414  ralxp  4422  ralrnmpt2  5557  fzprval  8714  fztpval  8715