Theorem eubii 2009

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 2008	. 2 ⊢ (⊤ → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
4	3	mptru 1341	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Syntax hints:   ↔ wb 104  ⊤wtru 1333  ∃!weu 2000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-eu 2003

This theorem is referenced by:  cbveu  2024  2eu7  2094  reubiia  2618  cbvreu  2655  reuv  2708  euxfr2dc  2873  euxfrdc  2874  2reuswapdc  2892  reuun2  3364  zfnuleu  4060  copsexg  4174  funeu2  5157  funcnv3  5193  fneu2  5236  tz6.12  5457  f1ompt  5579  fsn  5600  climreu  11098  divalgb  11658  txcn  12483