Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1345 Structured version   Visualization version   GIF version

Theorem bnj1345 32206
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1345.1 (𝜑 → ∃𝑥(𝜓𝜒))
bnj1345.2 (𝜃 ↔ (𝜑𝜓𝜒))
bnj1345.3 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
bnj1345 (𝜑 → ∃𝑥𝜃)

Proof of Theorem bnj1345
StepHypRef Expression
1 bnj1345.1 . . 3 (𝜑 → ∃𝑥(𝜓𝜒))
2 bnj1345.3 . . 3 (𝜑 → ∀𝑥𝜑)
31, 2bnj1275 32195 . 2 (𝜑 → ∃𝑥(𝜑𝜓𝜒))
4 bnj1345.2 . 2 (𝜃 ↔ (𝜑𝜓𝜒))
53, 4bnj1198 32177 1 (𝜑 → ∃𝑥𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀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  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786 This theorem is referenced by:  bnj1379  32212  bnj1521  32233
 Copyright terms: Public domain W3C validator