Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexlimddv2 Structured version   Visualization version   GIF version

Theorem rexlimddv2 40570
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 682 . 2 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
41, 3rexlimddv 3173 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-ral 3055  df-rex 3056
This theorem is referenced by:  climxlim2lem  40592
  Copyright terms: Public domain W3C validator