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Theorem gencbval 3283
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 3281 . . 3 (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ∃𝑦(𝜃 ∧ ¬ 𝜓))
7 exanali 1826 . . 3 (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜒𝜑))
8 exanali 1826 . . 3 (∃𝑦(𝜃 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦(𝜃𝜓))
96, 7, 83bitr3i 290 . 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 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-11 2074  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233
This theorem is referenced by: (None)
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