Theorem eubii 2035

Description: Introduce unique existential 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 2034	. 2 |- ( T. -> ( E! x ph <-> E! x ps ) )
4	3	mptru 1362	1 |- ( E! x ph <-> E! x ps )

Syntax hints:   <-> wb 105   T. wtru 1354   E!weu 2026

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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-eu 2029

This theorem is referenced by:  cbveu  2050  2eu7  2120  reubiia  2661  cbvreu  2701  reuv  2756  euxfr2dc  2922  euxfrdc  2923  2reuswapdc  2941  reuun2  3418  zfnuleu  4127  copsexg  4244  funeu2  5242  funcnv3  5278  fneu2  5321  tz6.12  5543  f1ompt  5667  fsn  5688  climreu  11304  divalgb  11929  txcn  13711