Theorem eubii 1906

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!xph <-> E!xps)

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 1905	. 2 |- ( T. -> (E!xph <-> E!xps))
4	3	trud 1251	1 |- (E!xph <-> E!xps)

Syntax hints:   <-> wb 98   T. wtru 1243  E!weu 1897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-eu 1900

This theorem is referenced by:  cbveu  1921  2eu7  1991  reubiia  2488  cbvreu  2525  reuv  2567  euxfr2dc  2720  euxfrdc  2721  2reuswapdc  2737  reuun2  3214  zfnuleu  3872  copsexg  3972  funeu2  4870  funcnv3  4904  fneu2  4947  tz6.12  5144  f1ompt  5263  fsn  5278