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

Theorem alexbii 1834
Description: Biconditional form of aleximi 1833. (Contributed by BJ, 16-Nov-2020.)
Hypothesis
Ref Expression
alexbii.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
alexbii (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Proof of Theorem alexbii
StepHypRef Expression
1 alexbii.1 . . . 4 (𝜑 → (𝜓𝜒))
21biimpd 232 . . 3 (𝜑 → (𝜓𝜒))
32aleximi 1833 . 2 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
41biimprd 251 . . 3 (𝜑 → (𝜒𝜓))
54aleximi 1833 . 2 (∀𝑥𝜑 → (∃𝑥𝜒 → ∃𝑥𝜓))
63, 5impbid 215 1 (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  exbi  1848  exbidh  1868  exintrbi  1892  eleq2d  2899  ralrexbid  3308  bnj956  32122  bj-2exbi  34023
  Copyright terms: Public domain W3C validator