MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfeu Structured version   Visualization version   GIF version

Theorem dfeu 2682
Description: Rederive df-eu 2655 from the old definition eu6 2660. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) (Proof shortened by BJ, 7-Oct-2022.) (Proof modification is discouraged.) Use df-eu 2655 instead. (New usage is discouraged.)
Assertion
Ref Expression
dfeu (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Proof of Theorem dfeu
StepHypRef Expression
1 abai 825 . 2 ((∃𝑥𝜑 ∧ ∃!𝑥𝜑) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃!𝑥𝜑)))
2 euex 2663 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
32pm4.71ri 564 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃!𝑥𝜑))
4 moeu 2669 . . 3 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
54anbi2i 625 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃!𝑥𝜑)))
61, 3, 53bitr4i 306 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  ∃*wmo 2622  ∃!weu 2654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2624  df-eu 2655
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator