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Theorem bnj596 31419
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 544 . 2 (𝜑 → (𝜑 ∧ ∃𝑥𝜓))
3 bnj596.1 . . . 4 (𝜑 → ∀𝑥𝜑)
43nf5i 2140 . . 3 𝑥𝜑
5419.42 2223 . 2 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
62, 5sylibr 226 1 (𝜑 → ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1599  wex 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828
This theorem is referenced by:  bnj1275  31487  bnj1340  31497  bnj594  31585  bnj1398  31705
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