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Theorem rexlimddv2 42633
 Description: Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
rexlimddv2.1 (𝜑 → ∃𝑥𝐴 𝜓)
rexlimddv2.2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
rexlimddv2 (𝜑𝜒)
Distinct variable groups:   𝜒,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimddv2
StepHypRef Expression
1 rexlimddv2.1 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
2 rexlimddv2.2 . . 3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
32anasss 470 . 2 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
41, 3rexlimddv 3251 1 (𝜑𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∃wrex 3107 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 1911 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3111  df-rex 3112 This theorem is referenced by:  climxlim2lem  42655  xlimliminflimsup  42672
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