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Theorem gencbval 3405
Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
Hypotheses
Ref Expression
gencbval.1 𝐴 ∈ V
gencbval.2 (𝐴 = 𝑦 → (𝜑𝜓))
gencbval.3 (𝐴 = 𝑦 → (𝜒𝜃))
gencbval.4 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
Assertion
Ref Expression
gencbval (∀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 gencbval
StepHypRef Expression
1 gencbval.1 . . . 4 𝐴 ∈ V
2 gencbval.2 . . . . 5 (𝐴 = 𝑦 → (𝜑𝜓))
32notbid 309 . . . 4 (𝐴 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
4 gencbval.3 . . . 4 (𝐴 = 𝑦 → (𝜒𝜃))
5 gencbval.4 . . . 4 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
61, 3, 4, 5gencbvex 3403 . . 3 (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ∃𝑦(𝜃 ∧ ¬ 𝜓))
7 exanali 1955 . . 3 (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜒𝜑))
8 exanali 1955 . . 3 (∃𝑦(𝜃 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦(𝜃𝜓))
96, 7, 83bitr3i 292 . 2 (¬ ∀𝑥(𝜒𝜑) ↔ ¬ ∀𝑦(𝜃𝜓))
109con4bii 312 1 (∀𝑥(𝜒𝜑) ↔ ∀𝑦(𝜃𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-v 3352
This theorem is referenced by: (None)
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