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Theorem gencbvex2 2618
Description: Restatement of gencbvex 2617 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 286 . . . 4 ((𝜒𝐴 = 𝑦) → 𝜃)
65exlimiv 1505 . . 3 (∃𝑥(𝜒𝐴 = 𝑦) → 𝜃)
74, 6impbii 121 . 2 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
81, 2, 3, 7gencbvex 2617 1 (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by: (None)
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