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Theorem bj-spcimdv 31874
Description: Remove from spcimdv 3262 dependency on ax-10 2005, ax-11 2020, ax-13 2233, ax-ext 2589, df-cleq 2602 (and df-nfc 2739, df-v 3174, df-tru 1477, df-nf 1700). (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 448 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
32alrimiv 1841 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 bj-spcimdv.1 . 2 (𝜑𝐴𝐵)
5 bj-elisset 31852 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
6 exim 1750 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓𝜒)))
75, 6syl5 33 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → ∃𝑥(𝜓𝜒)))
8 19.36v 1890 . . 3 (∃𝑥(𝜓𝜒) ↔ (∀𝑥𝜓𝜒))
97, 8syl6ib 239 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → (∀𝑥𝜓𝜒)))
103, 4, 9sylc 62 1 (𝜑 → (∀𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976
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-12 2033
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-sb 1867  df-clab 2596  df-clel 2605
This theorem is referenced by: (None)
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