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Theorem rexlimddv2 40567
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 457 . 2 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
41, 3rexlimddv 3183 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-ral 3066  df-rex 3067
This theorem is referenced by:  climxlim2lem  40589
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