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Theorem bnj596 32022
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 551 . 2 (𝜑 → (𝜑 ∧ ∃𝑥𝜓))
3 bnj596.1 . . . 4 (𝜑 → ∀𝑥𝜑)
43nf5i 2150 . . 3 𝑥𝜑
5419.42 2238 . 2 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
62, 5sylibr 236 1 (𝜑 → ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1535  wex 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785
This theorem is referenced by:  bnj1275  32090  bnj1340  32100  bnj594  32189  bnj1398  32311
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