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Theorem gencbval 3225
 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 307 . . . 4 (𝐴 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
4 gencbval.3 . . . 4 (𝐴 = 𝑦 → (𝜒𝜃))
5 gencbval.4 . . . 4 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
61, 3, 4, 5gencbvex 3223 . . 3 (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ∃𝑦(𝜃 ∧ ¬ 𝜓))
7 exanali 1773 . . 3 (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜒𝜑))
8 exanali 1773 . . 3 (∃𝑦(𝜃 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦(𝜃𝜓))
96, 7, 83bitr3i 289 . 2 (¬ ∀𝑥(𝜒𝜑) ↔ ¬ ∀𝑦(𝜃𝜓))
109con4bii 310 1 (∀𝑥(𝜒𝜑) ↔ ∀𝑦(𝜃𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by: (None)
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