Theorem eubii 2023

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 2022	. 2 ⊢ (⊤ → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
4	3	mptru 1352	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Syntax hints:   ↔ wb 104  ⊤wtru 1344  ∃!weu 2014

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-eu 2017

This theorem is referenced by:  cbveu  2038  2eu7  2108  reubiia  2650  cbvreu  2690  reuv  2745  euxfr2dc  2911  euxfrdc  2912  2reuswapdc  2930  reuun2  3405  zfnuleu  4106  copsexg  4222  funeu2  5214  funcnv3  5250  fneu2  5293  tz6.12  5514  f1ompt  5636  fsn  5657  climreu  11238  divalgb  11862  txcn  12915