Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1209 Structured version   Visualization version   GIF version

Theorem bnj1209 30121
 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 30119 . . . 4 (𝜒 → ∃𝑥(𝑥𝐵𝜑))
32ancli 572 . . 3 (𝜒 → (𝜒 ∧ ∃𝑥(𝑥𝐵𝜑)))
4 19.42v 1905 . . 3 (∃𝑥(𝜒 ∧ (𝑥𝐵𝜑)) ↔ (𝜒 ∧ ∃𝑥(𝑥𝐵𝜑)))
53, 4sylibr 223 . 2 (𝜒 → ∃𝑥(𝜒 ∧ (𝑥𝐵𝜑)))
6 bnj1209.2 . . 3 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
7 3anass 1035 . . 3 ((𝜒𝑥𝐵𝜑) ↔ (𝜒 ∧ (𝑥𝐵𝜑)))
86, 7bitri 263 . 2 (𝜃 ↔ (𝜒 ∧ (𝑥𝐵𝜑)))
95, 8bnj1198 30120 1 (𝜒 → ∃𝑥𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875 This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033  df-ex 1696  df-rex 2902 This theorem is referenced by:  bnj1501  30389  bnj1523  30393
 Copyright terms: Public domain W3C validator