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Theorem rexlimddvcbvw 42958
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 42957. The equivalent of this theorem without the bound variable change is rexlimddv 3162. Version of rexlimddvcbv 42959 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.)
Hypotheses
Ref Expression
rexlimddvcbvw.1 (𝜑 → ∃𝑥𝐴 𝜃)
rexlimddvcbvw.2 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
rexlimddvcbvw.3 (𝑥 = 𝑦 → (𝜃𝜒))
Assertion
Ref Expression
rexlimddvcbvw (𝜑𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑦   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)

Proof of Theorem rexlimddvcbvw
StepHypRef Expression
1 rexlimddvcbvw.1 . 2 (𝜑 → ∃𝑥𝐴 𝜃)
2 rexlimddvcbvw.3 . . . 4 (𝑥 = 𝑦 → (𝜃𝜒))
32cbvrexvw 3236 . . 3 (∃𝑥𝐴 𝜃 ↔ ∃𝑦𝐴 𝜒)
4 rexlimddvcbvw.2 . . . 4 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
54rexlimdvaa 3157 . . 3 (𝜑 → (∃𝑦𝐴 𝜒𝜓))
63, 5biimtrid 241 . 2 (𝜑 → (∃𝑥𝐴 𝜃𝜓))
71, 6mpd 15 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  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
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2811  df-rex 3072
This theorem is referenced by:  mnuprdlem1  43031  mnuprdlem2  43032
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