Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  gencbvex2 Structured version   Visualization version   GIF version

Theorem gencbvex2 3224
 Description: Restatement of gencbvex 3223 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
Hypotheses
Ref Expression
gencbvex2.1 𝐴 ∈ V
gencbvex2.2 (𝐴 = 𝑦 → (𝜑𝜓))
gencbvex2.3 (𝐴 = 𝑦 → (𝜒𝜃))
gencbvex2.4 (𝜃 → ∃𝑥(𝜒𝐴 = 𝑦))
Assertion
Ref Expression
gencbvex2 (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝜃,𝑥   𝜒,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝜃(𝑦)   𝐴(𝑥)

Proof of Theorem gencbvex2
StepHypRef Expression
1 gencbvex2.1 . 2 𝐴 ∈ V
2 gencbvex2.2 . 2 (𝐴 = 𝑦 → (𝜑𝜓))
3 gencbvex2.3 . 2 (𝐴 = 𝑦 → (𝜒𝜃))
4 gencbvex2.4 . . 3 (𝜃 → ∃𝑥(𝜒𝐴 = 𝑦))
53biimpac 502 . . . 4 ((𝜒𝐴 = 𝑦) → 𝜃)
65exlimiv 1845 . . 3 (∃𝑥(𝜒𝐴 = 𝑦) → 𝜃)
74, 6impbii 198 . 2 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
81, 2, 3, 7gencbvex 3223 1 (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173 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  ax-7 1922  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator