Theorem reu6i 3028

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 2362	. . . . 5 |- (y = B -> (x = y <-> x = B))
2	1	bibi2d 309	. . . 4 |- (y = B -> ((ph <-> x = y) <-> (ph <-> x = B)))
3	2	ralbidv 2635	. . 3 |- (y = B -> (A.x e. A (ph <-> x = y) <-> A.x e. A (ph <-> x = B)))
4	3	rspcev 2956	. 2 |- ((B e. A /\ A.x e. A (ph <-> x = B)) -> E.y e. A A.x e. A (ph <-> x = y))
5		reu6 3026	. 2 |- (E!x e. A ph <-> E.y e. A A.x e. A (ph <-> x = y))
6	4, 5	sylibr 203	1 |- ((B e. A /\ A.x e. A (ph <-> x = B)) -> E!x e. A ph)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2615  E.wrex 2616  E!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-reu 2622  df-v 2862

This theorem is referenced by:  eqreu  3029