Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exlimddvfi Structured version   Visualization version   GIF version

Theorem exlimddvfi 36979
Description: A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypotheses
Ref Expression
exlimddvfi.1 (𝜑 → ∃𝑥𝜃)
exlimddvfi.2 𝑦𝜃
exlimddvfi.3 𝑦𝜓
exlimddvfi.4 ([𝑦 / 𝑥]𝜃𝜂)
exlimddvfi.5 ((𝜂𝜓) → 𝜒)
exlimddvfi.6 𝑦𝜒
Assertion
Ref Expression
exlimddvfi ((𝜑𝜓) → 𝜒)

Proof of Theorem exlimddvfi
StepHypRef Expression
1 exlimddvfi.1 . . 3 (𝜑 → ∃𝑥𝜃)
2 exlimddvfi.2 . . . 4 𝑦𝜃
32sb8e 2518 . . 3 (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃)
41, 3sylib 217 . 2 (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃)
5 exlimddvfi.3 . 2 𝑦𝜓
6 sbsbc 3781 . . . 4 ([𝑦 / 𝑥]𝜃[𝑦 / 𝑥]𝜃)
7 exlimddvfi.4 . . . 4 ([𝑦 / 𝑥]𝜃𝜂)
86, 7bitri 275 . . 3 ([𝑦 / 𝑥]𝜃𝜂)
9 exlimddvfi.5 . . 3 ((𝜂𝜓) → 𝜒)
108, 9sylanb 582 . 2 (([𝑦 / 𝑥]𝜃𝜓) → 𝜒)
11 exlimddvfi.6 . 2 𝑦𝜒
124, 5, 10, 11exlimddvf 36978 1 ((𝜑𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wex 1782  wnf 1786  [wsb 2068  [wsbc 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2372  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sbc 3778
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator