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Theorem eubid 2487
Description: Formula-building rule for uniqueness quantifier (deduction rule). (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
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eubid.1 . . . 4 𝑥𝜑
2 eubid.2 . . . . 5 (𝜑 → (𝜓𝜒))
32bibi1d 333 . . . 4 (𝜑 → ((𝜓𝑥 = 𝑦) ↔ (𝜒𝑥 = 𝑦)))
41, 3albid 2088 . . 3 (𝜑 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜒𝑥 = 𝑦)))
54exbidv 1847 . 2 (𝜑 → (∃𝑦𝑥(𝜓𝑥 = 𝑦) ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦)))
6 df-eu 2473 . 2 (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
7 df-eu 2473 . 2 (∃!𝑥𝜒 ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦))
85, 6, 73bitr4g 303 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701  wnf 1705  ∃!weu 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707  df-eu 2473
This theorem is referenced by:  mobid  2488  eubidv  2489  euor  2511  euor2  2513  euan  2529  reubida  3113  reueq1f  3125  eusv2i  4825  reusv2lem3  4833  nbusgredgeu0  26164  eubi  38140
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