Theorem eubii 2088

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 2087	. 2 ⊢ (⊤ → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
4	3	mptru 1407	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Syntax hints:   ↔ wb 105  ⊤wtru 1399  ∃!weu 2079

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 2082

This theorem is referenced by:  cbveu  2103  2eu7  2174  reubiia  2720  cbvreu  2766  reuv  2823  euxfr2dc  2992  euxfrdc  2993  2reuswapdc  3011  reuun2  3492  zfnuleu  4218  copsexg  4342  funeu2  5359  funcnv3  5399  fneu2  5444  tz6.12  5676  f1ompt  5806  fsn  5827  climreu  11920  divalgb  12549  gsum0g  13542  txcn  15069