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Theorem bnj596 31148
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 530 . 2 (𝜑 → (𝜑 ∧ ∃𝑥𝜓))
3 bnj596.1 . . . 4 (𝜑 → ∀𝑥𝜑)
43nf5i 2178 . . 3 𝑥𝜑
5419.42-1 2259 . 2 ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
62, 5syl 17 1 (𝜑 → ∃𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857
This theorem is referenced by:  bnj1275  31216  bnj1340  31226  bnj594  31314  bnj1398  31434
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