MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.42 Structured version   Visualization version   GIF version

Theorem 19.42 2264
Description: Theorem 19.42 of [Margaris] p. 90. See 19.42v 2036 for a version requiring fewer axioms. See exan 1942 for an immediate version. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
19.42.1 𝑥𝜑
Assertion
Ref Expression
19.42 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Proof of Theorem 19.42
StepHypRef Expression
1 19.42.1 . . 3 𝑥𝜑
2119.41 2262 . 2 (∃𝑥(𝜓𝜑) ↔ (∃𝑥𝜓𝜑))
3 exancom 1941 . 2 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 ancom 449 . 2 ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓𝜑))
52, 3, 43bitr4i 293 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 383  wex 1855  wnf 1859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-12 2206
This theorem depends on definitions:  df-bi 198  df-an 384  df-ex 1856  df-nf 1861
This theorem is referenced by:  eean  2346  bnj916  31358  bnj983  31376
  Copyright terms: Public domain W3C validator