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Theorem cbvrexdva2 3346
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 318 . . . 4 ((𝜑𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒))
3 cbvraldva2.2 . . . 4 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
42, 3cbvraldva2 3345 . . 3 (𝜑 → (∀𝑥𝐴 ¬ 𝜓 ↔ ∀𝑦𝐵 ¬ 𝜒))
5 ralnex 3073 . . 3 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
6 ralnex 3073 . . 3 (∀𝑦𝐵 ¬ 𝜒 ↔ ¬ ∃𝑦𝐵 𝜒)
74, 5, 63bitr3g 313 . 2 (𝜑 → (¬ ∃𝑥𝐴 𝜓 ↔ ¬ ∃𝑦𝐵 𝜒))
87con4bid 317 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wral 3062  wrex 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072
This theorem is referenced by:  cbvrexdvaOLD  3349  mreexexlemd  17588  eulerpartlemgvv  33375  ismnu  43020
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