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  2662  cbvreu  2702  reuv  2757  euxfr2dc  2923  euxfrdc  2924  2reuswapdc  2942  reuun2  3419  zfnuleu  4128  copsexg  4245  funeu2  5243  funcnv3  5279  fneu2  5322  tz6.12  5544  f1ompt  5668  fsn  5689  climreu  11305  divalgb  11930  txcn  13778