Theorem reu6i 2964

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 2215	. . . . 5 |- ( y = B -> ( x = y <-> x = B ) )
2	1	bibi2d 232	. . . 4 |- ( y = B -> ( ( ph <-> x = y ) <-> ( ph <-> x = B ) ) )
3	2	ralbidv 2506	. . 3 |- ( y = B -> ( A. x e. A ( ph <-> x = y ) <-> A. x e. A ( ph <-> x = B ) ) )
4	3	rspcev 2877	. 2 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E. y e. A A. x e. A ( ph <-> x = y ) )
5		reu6 2962	. 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 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   E!wreu 2486

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-v 2774

This theorem is referenced by:  eqreu  2965  riota5f  5926  negeu  8265  creur  9034  creui  9035