Theorem reu6i 2998

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Assertion

Ref	Expression
reu6i	|- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem reu6i

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2241	. . . . 5 |- ( y = B -> ( x = y <-> x = B ) )
2	1	bibi2d 232	. . . 4 |- ( y = B -> ( ( ph <-> x = y ) <-> ( ph <-> x = B ) ) )
3	2	ralbidv 2533	. . 3 |- ( y = B -> ( A. x e. A ( ph <-> x = y ) <-> A. x e. A ( ph <-> x = B ) ) )
4	3	rspcev 2911	. 2 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E. y e. A A. x e. A ( ph <-> x = y ) )
5		reu6 2996	. 2 |- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) )
6	4, 5	sylibr 134	1 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   E!wreu 2513

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-v 2805

This theorem is referenced by:  eqreu  2999  riota5f  6008  negeu  8429  creur  9198  creui  9199  reuccatpfxs1  11394