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Mirrors > Home > MPE Home > Th. List > spcimdv | Structured version Visualization version GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid ax-10 2141 and ax-11 2156. (Revised by Gino Giotto, 20-Aug-2023.) |
Ref | Expression |
---|---|
spcimdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
spcimdv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
spcimdv | ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcimdv.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | elisset 3506 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
4 | spcimdv.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) | |
5 | 4 | ex 415 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒))) |
6 | 5 | eximdv 1914 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜒))) |
7 | 3, 6 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑥(𝜓 → 𝜒)) |
8 | 19.36v 1990 | . 2 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → 𝜒)) | |
9 | 7, 8 | sylib 220 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∃wex 1776 ∈ wcel 2110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-cleq 2814 df-clel 2893 |
This theorem is referenced by: spcdv 3593 spcimedv 3594 rspcimdv 3613 mrieqv2d 16904 mreexexlemd 16909 intabssd 39878 |
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