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Theorem rexlimdvaacbv 40830
 Description: Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3277. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypotheses
Ref Expression
rexlimdvaacbv.1 (𝑥 = 𝑦 → (𝜓𝜃))
rexlimdvaacbv.2 ((𝜑 ∧ (𝑦𝐴𝜃)) → 𝜒)
Assertion
Ref Expression
rexlimdvaacbv (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜓,𝑦   𝜃,𝑥   𝜑,𝑦   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)   𝜃(𝑦)

Proof of Theorem rexlimdvaacbv
StepHypRef Expression
1 rexlimdvaacbv.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜃))
21cbvrexv 3438 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜃)
3 rexlimdvaacbv.2 . . 3 ((𝜑 ∧ (𝑦𝐴𝜃)) → 𝜒)
43rexlimdvaa 3277 . 2 (𝜑 → (∃𝑦𝐴 𝜃𝜒))
52, 4syl5bi 245 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2115  ∃wrex 3134 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139 This theorem is referenced by:  rexlimddvcbv  40832
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