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Theorem exlimddvfi 35518
 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 2560 . . 3 (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃)
41, 3sylib 221 . 2 (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃)
5 exlimddvfi.3 . 2 𝑦𝜓
6 sbsbc 3751 . . . 4 ([𝑦 / 𝑥]𝜃[𝑦 / 𝑥]𝜃)
7 exlimddvfi.4 . . . 4 ([𝑦 / 𝑥]𝜃𝜂)
86, 7bitri 278 . . 3 ([𝑦 / 𝑥]𝜃𝜂)
9 exlimddvfi.5 . . 3 ((𝜂𝜓) → 𝜒)
108, 9sylanb 584 . 2 (([𝑦 / 𝑥]𝜃𝜓) → 𝜒)
11 exlimddvfi.6 . 2 𝑦𝜒
124, 5, 10, 11exlimddvf 35517 1 ((𝜑𝜓) → 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1781  Ⅎwnf 1785  [wsb 2069  [wsbc 3747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-13 2391  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-sbc 3748 This theorem is referenced by: (None)
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