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Theorem bnj1209 29927
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1209.1 (𝜒 → ∃𝑥𝐵 𝜑)
bnj1209.2 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
Assertion
Ref Expression
bnj1209 (𝜒 → ∃𝑥𝜃)
Distinct variable group:   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜃(𝑥)   𝐵(𝑥)

Proof of Theorem bnj1209
StepHypRef Expression
1 bnj1209.1 . . . . 5 (𝜒 → ∃𝑥𝐵 𝜑)
21bnj1196 29925 . . . 4 (𝜒 → ∃𝑥(𝑥𝐵𝜑))
32ancli 571 . . 3 (𝜒 → (𝜒 ∧ ∃𝑥(𝑥𝐵𝜑)))
4 19.42v 1904 . . 3 (∃𝑥(𝜒 ∧ (𝑥𝐵𝜑)) ↔ (𝜒 ∧ ∃𝑥(𝑥𝐵𝜑)))
53, 4sylibr 222 . 2 (𝜒 → ∃𝑥(𝜒 ∧ (𝑥𝐵𝜑)))
6 bnj1209.2 . . 3 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
7 3anass 1034 . . 3 ((𝜒𝑥𝐵𝜑) ↔ (𝜒 ∧ (𝑥𝐵𝜑)))
86, 7bitri 262 . 2 (𝜃 ↔ (𝜒 ∧ (𝑥𝐵𝜑)))
95, 8bnj1198 29926 1 (𝜒 → ∃𝑥𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030  wex 1694  wcel 1976  wrex 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874
This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-ex 1695  df-rex 2901
This theorem is referenced by:  bnj1501  30195  bnj1523  30199
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