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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-spcimdvv | Structured version Visualization version GIF version |
Description: Remove from spcimdv 3589 dependency on ax-7 2006, ax-8 2107, ax-10 2136, ax-11 2151, ax-12 2167 ax-13 2381, ax-ext 2790, df-cleq 2811, df-clab 2797 (and df-nfc 2960, df-v 3494, df-or 842, df-tru 1531, df-nf 1776) 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 34108. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-spcimdvv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
bj-spcimdvv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
bj-spcimdvv | ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-spcimdvv.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) | |
2 | 1 | ex 413 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒))) |
3 | 2 | alrimiv 1919 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒))) |
4 | bj-spcimdvv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
5 | bj-elissetv 34088 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
6 | exim 1825 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜒))) | |
7 | 5, 6 | syl5 34 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → ∃𝑥(𝜓 → 𝜒))) |
8 | 19.36v 1985 | . . 3 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → 𝜒)) | |
9 | 7, 8 | syl6ib 252 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜓 → 𝜒))) |
10 | 3, 4, 9 | sylc 65 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-clel 2890 |
This theorem is referenced by: (None) |
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