Theorem reueqd 2742

Description: Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)

Hypothesis

Ref	Expression
raleqd.1	|- ( A = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
reueqd	|- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reueqd

Step	Hyp	Ref	Expression
1		reueq1 2730	. 2 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	reubidv 2716	. 2 |- ( A = B -> ( E! x e. B ph <-> E! x e. B ps ) )
4	1, 3	bitrd 188	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   E!wreu 2510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-cleq 2222  df-clel 2225  df-nfc 2361  df-reu 2515

This theorem is referenced by:  acexmidlemcase  5996  srgideu  13935