Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ceqsalg0 Structured version   Visualization version   GIF version

Theorem bj-ceqsalg0 32861
 Description: The FOL content of ceqsalg 3228. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-ceqsalg0.1 𝑥𝜓
bj-ceqsalg0.2 (𝜒 → (𝜑𝜓))
Assertion
Ref Expression
bj-ceqsalg0 (∃𝑥𝜒 → (∀𝑥(𝜒𝜑) ↔ 𝜓))

Proof of Theorem bj-ceqsalg0
StepHypRef Expression
1 bj-ceqsalg0.1 . 2 𝑥𝜓
2 bj-ceqsalg0.2 . . 3 (𝜒 → (𝜑𝜓))
32ax-gen 1721 . 2 𝑥(𝜒 → (𝜑𝜓))
4 bj-ceqsalt0 32857 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ ∃𝑥𝜒) → (∀𝑥(𝜒𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1413 1 (∃𝑥𝜒 → (∀𝑥(𝜒𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1480  ∃wex 1703  Ⅎwnf 1707 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-ex 1704  df-nf 1709 This theorem is referenced by:  bj-ceqsalg  32862  bj-ceqsalgv  32864
 Copyright terms: Public domain W3C validator