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Theorem rexlimddvcbvw 44196
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44195. The equivalent of this theorem without the bound variable change is rexlimddv 3159. Version of rexlimddvcbv 44197 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by GG, 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 3154 . . 3 (𝜑 → (∃𝑦𝐴 𝜒𝜓))
63, 5biimtrid 242 . 2 (𝜑 → (∃𝑥𝐴 𝜃𝜓))
71, 6mpd 15 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-clel 2814  df-rex 3069
This theorem is referenced by:  mnuprdlem1  44268  mnuprdlem2  44269
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