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Theorem gencbvex2 3471
Description: Restatement of gencbvex 3470 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 471 . . . 4 ((𝜒𝐴 = 𝑦) → 𝜃)
65exlimiv 1889 . . 3 (∃𝑥(𝜒𝐴 = 𝑦) → 𝜃)
74, 6impbii 201 . 2 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
81, 2, 3, 7gencbvex 3470 1 (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wex 1742  wcel 2050  Vcvv 3415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-cleq 2771  df-clel 2846
This theorem is referenced by: (None)
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