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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 𝑥𝜑
eubid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
eubid (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Proof of Theorem eubid
StepHypRef Expression
1 eubid.1 . . . 4 𝑥𝜑
2 eubid.2 . . . 4 (𝜑 → (𝜓𝜒))
31, 2exbid 2258 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
41, 2mobid 2607 . . 3 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
53, 4anbi12d 625 . 2 (𝜑 → ((∃𝑥𝜓 ∧ ∃*𝑥𝜓) ↔ (∃𝑥𝜒 ∧ ∃*𝑥𝜒)))
6 df-eu 2609 . 2 (∃!𝑥𝜓 ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓))
7 df-eu 2609 . 2 (∃!𝑥𝜒 ↔ (∃𝑥𝜒 ∧ ∃*𝑥𝜒))
85, 6, 73bitr4g 306 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wex 1875  wnf 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
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