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Theorem cbvrexdva 3240
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Avoid ax-9 2118, ax-ext 2708. (Revised by Wolf Lammen, 9-Mar-2025.)
Hypothesis
Ref Expression
cbvraldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvrexdva (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvrexdva
StepHypRef Expression
1 cbvraldva.1 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
21notbid 318 . . . 4 ((𝜑𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒))
32cbvraldva 3239 . . 3 (𝜑 → (∀𝑥𝐴 ¬ 𝜓 ↔ ∀𝑦𝐴 ¬ 𝜒))
4 ralnex 3072 . . 3 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
5 ralnex 3072 . . 3 (∀𝑦𝐴 ¬ 𝜒 ↔ ¬ ∃𝑦𝐴 𝜒)
63, 4, 53bitr3g 313 . 2 (𝜑 → (¬ ∃𝑥𝐴 𝜓 ↔ ¬ ∃𝑦𝐴 𝜒))
76con4bid 317 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wral 3061  wrex 3070
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 1910  ax-6 1967  ax-7 2007  ax-8 2110
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-clel 2816  df-ral 3062  df-rex 3071
This theorem is referenced by:  tfrlem3a  8417  2sqmo  27481  trgcopy  28812  trgcopyeu  28814  acopyeu  28842  tgasa1  28866  dispcmp  33858  satffunlem1lem1  35407  satffunlem2lem1  35409  cbviundavw  36263  f1omptsn  37338  pibt2  37418  prjsprel  42614  opnneilem  48803
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