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Theorem exlimddvfi 32880
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 2412 . . 3 (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃)
41, 3sylib 206 . 2 (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃)
5 exlimddvfi.3 . 2 𝑦𝜓
6 sbsbc 3405 . . . 4 ([𝑦 / 𝑥]𝜃[𝑦 / 𝑥]𝜃)
7 exlimddvfi.4 . . . 4 ([𝑦 / 𝑥]𝜃𝜂)
86, 7bitri 262 . . 3 ([𝑦 / 𝑥]𝜃𝜂)
9 exlimddvfi.5 . . 3 ((𝜂𝜓) → 𝜒)
108, 9sylanb 487 . 2 (([𝑦 / 𝑥]𝜃𝜓) → 𝜒)
11 exlimddvfi.6 . 2 𝑦𝜒
124, 5, 10, 11exlimddvf 32879 1 ((𝜑𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wex 1694  wnf 1698  [wsb 1866  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-sbc 3402
This theorem is referenced by: (None)
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