Theorem eubii 2095

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	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
eubii	⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Proof of Theorem eubii

Step	Hyp	Ref	Expression
1		eubii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	a1i 9	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3	2	eubidv 2094	. 2 ⊢ (⊤ → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
4	3	mptru 1411	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Syntax hints:   ↔ wb 105  ⊤wtru 1403  ∃!weu 2086

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-eu 2089

This theorem is referenced by:  cbveu  2110  2eu7  2181  reubiia  2738  cbvreu  2784  reuv  2841  euxfr2dc  3011  euxfrdc  3012  2reuswapdc  3030  reuun2  3516  zfnuleu  4252  copsexg  4379  funeu2  5398  funcnv3  5438  fneu2  5483  tz6.12  5718  f1ompt  5850  fsn  5871  climreu  12041  divalgb  12670  gzsum0  13690  txcn  15299