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Theorem bj-spcimdvv 34651
 Description: Remove from spcimdv 3512 dependency on ax-7 2015, ax-8 2113, ax-10 2142, ax-11 2158, ax-12 2175 ax-13 2379, ax-ext 2729, df-cleq 2750, df-clab 2736 (and df-nfc 2901, df-v 3411, df-or 845, df-tru 1541, df-nf 1786) at the price of adding a disjoint variable condition on 𝑥, 𝐵 (but in usages, 𝑥 is typically a dummy, hence fresh, variable). For the version without this disjoint variable condition, see bj-spcimdv 34650. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-spcimdvv.1 (𝜑𝐴𝐵)
bj-spcimdvv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
bj-spcimdvv (𝜑 → (∀𝑥𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem bj-spcimdvv
StepHypRef Expression
1 bj-spcimdvv.2 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
21ex 416 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
32alrimiv 1928 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 bj-spcimdvv.1 . 2 (𝜑𝐴𝐵)
5 bj-elissetv 34630 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
6 exim 1835 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓𝜒)))
75, 6syl5 34 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → ∃𝑥(𝜓𝜒)))
8 19.36v 1994 . . 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 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2830 This theorem is referenced by: (None)
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