MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.43 Structured version   Visualization version   GIF version

Theorem r19.43 3123
Description: Restricted quantifier version of 19.43 1886. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.43 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Proof of Theorem r19.43
StepHypRef Expression
1 r19.35 3109 . 2 (∃𝑥𝐴𝜑𝜓) ↔ (∀𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 𝜓))
2 df-or 847 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32rexbii 3095 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥𝐴𝜑𝜓))
4 df-or 847 . . 3 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (¬ ∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
5 ralnex 3073 . . . 4 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
65imbi1i 350 . . 3 ((∀𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 𝜓) ↔ (¬ ∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
74, 6bitr4i 278 . 2 ((∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 ¬ 𝜑 → ∃𝑥𝐴 𝜓))
81, 3, 73bitr4i 303 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 846  wral 3062  wrex 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-ral 3063  df-rex 3072
This theorem is referenced by:  r19.45v  3193  r19.44v  3194  r19.45zv  4503  r19.44zv  4504  iunun  5097  soseq  8145  wemapsolem  9545  pythagtriplem2  16750  pythagtrip  16767  dcubic  26351  addsdilem1  27606  mulsasslem2  27619  legtrid  27842  axcontlem4  28225  erdszelem11  34192  satfvsucsuc  34356  fmla1  34378  seglelin  35088  diophun  41511  rexzrexnn0  41542  ldepslinc  47190
  Copyright terms: Public domain W3C validator