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Theorem bj-spcimdvv 32585
Description: Remove from spcimdv 3280 dependency on ax-7 1932, ax-8 1989, ax-10 2016, ax-11 2031, ax-12 2044 ax-13 2245, ax-ext 2601, df-cleq 2614, df-clab 2608 (and df-nfc 2750, df-v 3192, df-or 385, df-tru 1483, df-nf 1707) at the price of adding a DV condition on 𝑥, 𝐵 (but in usages, 𝑥 is typically a dummy, hence fresh, variable). For the version without this DV condition, see bj-spcimdv 32584. (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 450 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
32alrimiv 1852 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 bj-spcimdvv.1 . 2 (𝜑𝐴𝐵)
5 bj-elissetv 32561 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
6 exim 1758 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓𝜒)))
75, 6syl5 34 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → ∃𝑥(𝜓𝜒)))
8 19.36v 1901 . . 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 384  wal 1478   = wceq 1480  wex 1701  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-clel 2617
This theorem is referenced by: (None)
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