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Theorem cbvexdvaw 2046
 Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 2420 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024.)
Hypothesis
Ref Expression
cbvaldvaw.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvexdvaw (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvexdvaw
StepHypRef Expression
1 cbvaldvaw.1 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
21notbid 321 . . . 4 ((𝜑𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒))
32cbvaldvaw 2045 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
4 alnex 1783 . . 3 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5 alnex 1783 . . 3 (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒)
63, 4, 53bitr3g 316 . 2 (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒))
76con4bid 320 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  cbvex2vw  2048  isinf  8733  grumnud  41165
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