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

Theorem euanv 2627
Description: Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)
Assertion
Ref Expression
euanv (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem euanv
StepHypRef Expression
1 euex 2578 . . . 4 (∃!𝑥(𝜑𝜓) → ∃𝑥(𝜑𝜓))
2 simpl 486 . . . . 5 ((𝜑𝜓) → 𝜑)
32exlimiv 1936 . . . 4 (∃𝑥(𝜑𝜓) → 𝜑)
41, 3syl 17 . . 3 (∃!𝑥(𝜑𝜓) → 𝜑)
5 ibar 532 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
65eubidv 2587 . . . 4 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
76biimprcd 253 . . 3 (∃!𝑥(𝜑𝜓) → (𝜑 → ∃!𝑥𝜓))
84, 7jcai 520 . 2 (∃!𝑥(𝜑𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
96biimpa 480 . 2 ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
108, 9impbii 212 1 (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1786  ∃!weu 2569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-mo 2540  df-eu 2570
This theorem is referenced by:  eueq2  3607  2reu5lem1  3652  fsn  6901  dfac5lem5  9620
  Copyright terms: Public domain W3C validator