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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 2144 and ax-11 2160. (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 3508 | . . . 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 1917 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜒))) |
7 | 3, 6 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑥(𝜓 → 𝜒)) |
8 | 19.36v 1993 | . 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 1534 = wceq 1536 ∃wex 1779 ∈ wcel 2113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-cleq 2817 df-clel 2896 |
This theorem is referenced by: spcdv 3596 spcimedv 3597 rspcimdv 3616 mrieqv2d 16913 mreexexlemd 16918 intabssd 39891 |
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