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Theorem gencbvex2 3470
 Description: Restatement of gencbvex 3469 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 482 . . . 4 ((𝜒𝐴 = 𝑦) → 𝜃)
65exlimiv 1932 . . 3 (∃𝑥(𝜒𝐴 = 𝑦) → 𝜃)
74, 6impbii 212 . 2 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
81, 2, 3, 7gencbvex 3469 1 (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1539  ∃wex 1782   ∈ wcel 2112  Vcvv 3410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1783  df-nf 1787  df-cleq 2751  df-clel 2831 This theorem is referenced by: (None)
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