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Theorem bj-spcimdv 34652
 Description: Remove from spcimdv 3513 dependency on ax-9 2122, ax-10 2143, ax-11 2159, ax-13 2380, ax-ext 2730, df-cleq 2751 (and df-nfc 2902, df-v 3412, df-or 845, df-tru 1542, df-nf 1787). For an even more economical version, see bj-spcimdvv 34653. (Contributed by BJ, 30-Nov-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-spcimdv.1 (𝜑𝐴𝐵)
bj-spcimdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
bj-spcimdv (𝜑 → (∀𝑥𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem bj-spcimdv
StepHypRef Expression
1 bj-spcimdv.2 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
21ex 416 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
32alrimiv 1929 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 bj-spcimdv.1 . 2 (𝜑𝐴𝐵)
5 bj-elisset 34633 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
6 exim 1836 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓𝜒)))
75, 6syl5 34 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → ∃𝑥(𝜓𝜒)))
8 19.36v 1995 . . 3 (∃𝑥(𝜓𝜒) ↔ (∀𝑥𝜓𝜒))
97, 8syl6ib 254 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → (∀𝑥𝜓𝜒)))
103, 4, 9sylc 65 1 (𝜑 → (∀𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112 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 1912  ax-6 1971  ax-7 2016  ax-8 2114 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-clel 2831 This theorem is referenced by: (None)
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