Theorem eubii 1950

Description: Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
eubii.1	|- ( ph <-> ps )

Assertion

Ref	Expression
eubii	|- ( E! x ph <-> E! x ps )

Proof of Theorem eubii

Step	Hyp	Ref	Expression
1		eubii.1	. . . 4 |- ( ph <-> ps )
2	1	a1i 9	. . 3 |- ( T. -> ( ph <-> ps ) )
3	2	eubidv 1949	. 2 |- ( T. -> ( E! x ph <-> E! x ps ) )
4	3	trud 1293	1 |- ( E! x ph <-> E! x ps )

Syntax hints:   <-> wb 103   T. wtru 1285   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-eu 1944

This theorem is referenced by:  cbveu  1965  2eu7  2035  reubiia  2538  cbvreu  2575  reuv  2618  euxfr2dc  2777  euxfrdc  2778  2reuswapdc  2794  reuun2  3247  zfnuleu  3902  copsexg  3999  funeu2  4947  funcnv3  4981  fneu2  5024  tz6.12  5222  f1ompt  5341  fsn  5356  climreu  10136  divalgb  10325