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Theorem bnj1209 31384
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 31382 . . . 4 (𝜒 → ∃𝑥(𝑥𝐵𝜑))
32ancli 545 . . 3 (𝜒 → (𝜒 ∧ ∃𝑥(𝑥𝐵𝜑)))
4 19.42v 2049 . . 3 (∃𝑥(𝜒 ∧ (𝑥𝐵𝜑)) ↔ (𝜒 ∧ ∃𝑥(𝑥𝐵𝜑)))
53, 4sylibr 226 . 2 (𝜒 → ∃𝑥(𝜒 ∧ (𝑥𝐵𝜑)))
6 bnj1209.2 . . 3 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
7 3anass 1117 . . 3 ((𝜒𝑥𝐵𝜑) ↔ (𝜒 ∧ (𝑥𝐵𝜑)))
86, 7bitri 267 . 2 (𝜃 ↔ (𝜒 ∧ (𝑥𝐵𝜑)))
95, 8bnj1198 31383 1 (𝜒 → ∃𝑥𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wex 1875  wcel 2157  wrex 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006
This theorem depends on definitions:  df-bi 199  df-an 386  df-3an 1110  df-ex 1876  df-rex 3095
This theorem is referenced by:  bnj1501  31652  bnj1523  31656
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