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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-spcimdv | Structured version Visualization version GIF version |
Description: Remove from spcimdv 3552 dependency on ax-9 2116, ax-10 2137, ax-11 2154, ax-13 2370, ax-ext 2707, df-cleq 2728 (and df-nfc 2889, df-v 3447, df-or 846, df-tru 1544, df-nf 1786). For an even more economical version, see bj-spcimdvv 35363. (Contributed by BJ, 30-Nov-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-spcimdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
bj-spcimdv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
bj-spcimdv | ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-spcimdv.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) | |
2 | 1 | ex 413 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒))) |
3 | 2 | alrimiv 1930 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒))) |
4 | bj-spcimdv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
5 | elisset 2819 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
6 | exim 1836 | . . . 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 1991 | . . 3 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → 𝜒)) | |
9 | 7, 8 | syl6ib 250 | . 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 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-clel 2814 |
This theorem is referenced by: (None) |
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