Theorem reueqd 2683

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 2675	. 2 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	reubidv 2661	. 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 1353   E!wreu 2457

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-cleq 2170  df-clel 2173  df-nfc 2308  df-reu 2462

This theorem is referenced by:  acexmidlemcase  5870  srgideu  13155