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Theorem exlimddvfi 37294
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 2516 . . 3 (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃)
41, 3sylib 217 . 2 (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃)
5 exlimddvfi.3 . 2 𝑦𝜓
6 sbsbc 3782 . . . 4 ([𝑦 / 𝑥]𝜃[𝑦 / 𝑥]𝜃)
7 exlimddvfi.4 . . . 4 ([𝑦 / 𝑥]𝜃𝜂)
86, 7bitri 274 . . 3 ([𝑦 / 𝑥]𝜃𝜂)
9 exlimddvfi.5 . . 3 ((𝜂𝜓) → 𝜒)
108, 9sylanb 580 . 2 (([𝑦 / 𝑥]𝜃𝜓) → 𝜒)
11 exlimddvfi.6 . 2 𝑦𝜒
124, 5, 10, 11exlimddvf 37293 1 ((𝜑𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wex 1780  wnf 1784  [wsb 2066  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-sbc 3779
This theorem is referenced by: (None)
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