Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-spcimdv Structured version   Visualization version   GIF version

Theorem bj-spcimdv 32078
 Description: Remove from spcimdv 3263 dependency on ax-10 2006, ax-11 2021, ax-13 2234, ax-ext 2590, df-cleq 2603 (and df-nfc 2740, df-v 3175, df-tru 1478, df-nf 1701). (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 449 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
32alrimiv 1842 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 bj-spcimdv.1 . 2 (𝜑𝐴𝐵)
5 bj-elisset 32056 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
6 exim 1751 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓𝜒)))
75, 6syl5 33 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → ∃𝑥(𝜓𝜒)))
8 19.36v 1891 . . 3 (∃𝑥(𝜓𝜒) ↔ (∀𝑥𝜓𝜒))
97, 8syl6ib 240 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → (∀𝑥𝜓𝜒)))
103, 4, 9sylc 63 1 (𝜑 → (∀𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-clel 2606 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator