Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexlimd3 Structured version   Visualization version   GIF version

Theorem rexlimd3 41289
Description: * Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
rexlimd3.1 𝑥𝜑
rexlimd3.2 𝑥𝜒
rexlimd3.3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
rexlimd3 (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Proof of Theorem rexlimd3
StepHypRef Expression
1 rexlimd3.1 . 2 𝑥𝜑
2 rexlimd3.2 . 2 𝑥𝜒
3 rexlimd3.3 . . 3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
43exp31 420 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
51, 2, 4rexlimd 3314 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wnf 1775  wcel 2105  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-ral 3140  df-rex 3141
This theorem is referenced by:  dffo3f  41314
  Copyright terms: Public domain W3C validator