Theorem eubid 2627

Description: Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)

Hypotheses

Ref	Expression
eubid.1	⊢ Ⅎ𝑥𝜑
eubid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
eubid	⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Proof of Theorem eubid

Step	Hyp	Ref	Expression
1		eubid.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eubid.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	exbid 2258	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
4	1, 2	mobid 2607	. . 3 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
5	3, 4	anbi12d 625	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 ∧ ∃*𝑥𝜓) ↔ (∃𝑥𝜒 ∧ ∃*𝑥𝜒)))
6		df-eu 2609	. 2 ⊢ (∃!𝑥𝜓 ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓))
7		df-eu 2609	. 2 ⊢ (∃!𝑥𝜒 ↔ (∃𝑥𝜒 ∧ ∃*𝑥𝜒))
8	5, 6, 7	3bitr4g 306	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385  ∃wex 1875  Ⅎwnf 1879  ∃*wmo 2589  ∃!weu 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880  df-mo 2591  df-eu 2609

This theorem is referenced by:  eubidvOLDOLD  2643  mobidOLD  2646  euor  2663  euor2  2666  euan  2684  reubida  3305  reueq1f  3318  eusv2i  5063  reusv2lem3  5069  eubiOLD  39407