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Theorem cbvrexdva2 3343
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 8-Jan-2025.)
Hypotheses
Ref Expression
cbvraldva2.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
cbvraldva2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvrexdva2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Distinct variable groups:   𝑦,𝐴   𝜓,𝑦   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrexdva2
StepHypRef Expression
1 cbvraldva2.1 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
21notbid 317 . . . 4 ((𝜑𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒))
3 cbvraldva2.2 . . . 4 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
42, 3cbvraldva2 3342 . . 3 (𝜑 → (∀𝑥𝐴 ¬ 𝜓 ↔ ∀𝑦𝐵 ¬ 𝜒))
5 ralnex 3070 . . 3 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
6 ralnex 3070 . . 3 (∀𝑦𝐵 ¬ 𝜒 ↔ ¬ ∃𝑦𝐵 𝜒)
74, 5, 63bitr3g 312 . 2 (𝜑 → (¬ ∃𝑥𝐴 𝜓 ↔ ¬ ∃𝑦𝐵 𝜒))
87con4bid 316 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1539  wral 3059  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069
This theorem is referenced by:  cbvrexdvaOLD  3346  mreexexlemd  17592  eulerpartlemgvv  33673  ismnu  43322
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