MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.45 Structured version   Visualization version   GIF version

Theorem 19.45 2092
Description: Theorem 19.45 of [Margaris] p. 90. See 19.45v 1899 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
Hypothesis
Ref Expression
19.45.1 𝑥𝜑
Assertion
Ref Expression
19.45 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.45
StepHypRef Expression
1 19.43 1798 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.45.1 . . . 4 𝑥𝜑
3219.9 2058 . . 3 (∃𝑥𝜑𝜑)
43orbi1i 540 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
51, 4bitri 262 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wo 381  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695  df-nf 1700
This theorem is referenced by:  eeor  2155
  Copyright terms: Public domain W3C validator