Theorem eubii 2089

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 2088	. 2 ⊢ (⊤ → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
4	3	mptru 1407	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Syntax hints:   ↔ wb 105  ⊤wtru 1399  ∃!weu 2080

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-eu 2083

This theorem is referenced by:  cbveu  2104  2eu7  2175  reubiia  2730  cbvreu  2776  reuv  2833  euxfr2dc  3002  euxfrdc  3003  2reuswapdc  3021  reuun2  3504  zfnuleu  4234  copsexg  4360  funeu2  5378  funcnv3  5418  fneu2  5463  tz6.12  5698  f1ompt  5828  fsn  5849  climreu  11982  divalgb  12611  gsum0g  13609  txcn  15140