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Theorem gencbval 3490
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 318 . . . 4 (𝐴 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
4 gencbval.3 . . . 4 (𝐴 = 𝑦 → (𝜒𝜃))
5 gencbval.4 . . . 4 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
61, 3, 4, 5gencbvex 3488 . . 3 (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ∃𝑦(𝜃 ∧ ¬ 𝜓))
7 exanali 1862 . . 3 (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜒𝜑))
8 exanali 1862 . . 3 (∃𝑦(𝜃 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦(𝜃𝜓))
96, 7, 83bitr3i 301 . 2 (¬ ∀𝑥(𝜒𝜑) ↔ ¬ ∀𝑦(𝜃𝜓))
109con4bii 321 1 (∀𝑥(𝜒𝜑) ↔ ∀𝑦(𝜃𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1537   = wceq 1539  wex 1782  wcel 2106  Vcvv 3432
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-clel 2816
This theorem is referenced by: (None)
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