Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj596 Structured version   Visualization version   GIF version

Theorem bnj596 31133
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj596.1 (𝜑 → ∀𝑥𝜑)
bnj596.2 (𝜑 → ∃𝑥𝜓)
Assertion
Ref Expression
bnj596 (𝜑 → ∃𝑥(𝜑𝜓))

Proof of Theorem bnj596
StepHypRef Expression
1 bnj596.2 . . 3 (𝜑 → ∃𝑥𝜓)
21ancli 540 . 2 (𝜑 → (𝜑 ∧ ∃𝑥𝜓))
3 bnj596.1 . . . 4 (𝜑 → ∀𝑥𝜑)
43nf5i 2189 . . 3 𝑥𝜑
5419.42-1 2271 . 2 ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
62, 5syl 17 1 (𝜑 → ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635  wex 1859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-10 2184  ax-12 2213
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-nf 1864
This theorem is referenced by:  bnj1275  31201  bnj1340  31211  bnj594  31299  bnj1398  31419
  Copyright terms: Public domain W3C validator