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Theorem exlimddvfi 33097
 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 2413 . . 3 (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃)
41, 3sylib 207 . 2 (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃)
5 exlimddvfi.3 . 2 𝑦𝜓
6 sbsbc 3406 . . . 4 ([𝑦 / 𝑥]𝜃[𝑦 / 𝑥]𝜃)
7 exlimddvfi.4 . . . 4 ([𝑦 / 𝑥]𝜃𝜂)
86, 7bitri 263 . . 3 ([𝑦 / 𝑥]𝜃𝜂)
9 exlimddvfi.5 . . 3 ((𝜂𝜓) → 𝜒)
108, 9sylanb 488 . 2 (([𝑦 / 𝑥]𝜃𝜓) → 𝜒)
11 exlimddvfi.6 . 2 𝑦𝜒
124, 5, 10, 11exlimddvf 33096 1 ((𝜑𝜓) → 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695  Ⅎwnf 1699  [wsb 1867  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403 This theorem is referenced by: (None)
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