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Theorem bj-spcimdv 33213
Description: Remove from spcimdv 3441 dependency on ax-9 2154, ax-10 2174, ax-11 2190, ax-13 2408, ax-ext 2751, df-cleq 2764 (and df-nfc 2902, df-v 3353, df-or 835, df-tru 1634, df-nf 1858). For an even more economical version, see bj-spcimdvv 33214. (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 397 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
32alrimiv 2007 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 bj-spcimdv.1 . 2 (𝜑𝐴𝐵)
5 bj-elisset 33191 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
6 exim 1909 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓𝜒)))
75, 6syl5 34 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → ∃𝑥(𝜓𝜒)))
8 19.36v 2072 . . 3 (∃𝑥(𝜓𝜒) ↔ (∀𝑥𝜓𝜒))
97, 8syl6ib 241 . 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 382  wal 1629   = wceq 1631  wex 1852  wcel 2145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-sb 2050  df-clab 2758  df-clel 2767
This theorem is referenced by: (None)
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