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Theorem cbvexdva 2416
Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2156. (Revised by Wolf Lammen, 18-Jul-2021.)
Hypothesis
Ref Expression
cbvaldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvexdva (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvexdva
StepHypRef Expression
1 cbvaldva.1 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
21notbid 307 . . . 4 ((𝜑𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒))
32cbvaldva 2414 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
4 alnex 1843 . . 3 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5 alnex 1843 . . 3 (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒)
63, 4, 53bitr3g 302 . 2 (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒))
76con4bid 306 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1618  wex 1841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-11 2171  ax-12 2184  ax-13 2379
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1842
This theorem is referenced by:  cbvex2v  2420  cbvrexdva2  3303  isinf  8326
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