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Mirrors > Home > MPE Home > Th. List > rspcimdv | Structured version Visualization version GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rspcimdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcimdv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
rspcimdv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3087 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
2 | rspcimdv.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | simpr 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
4 | 3 | eleq1d 2844 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
5 | 4 | biimprd 240 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐵)) |
6 | rspcimdv.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) | |
7 | 5, 6 | imim12d 81 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒))) |
8 | 2, 7 | spcimdv 3505 | . . 3 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒))) |
9 | 2, 8 | mpid 44 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → 𝜒)) |
10 | 1, 9 | syl5bi 234 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∀wal 1505 = wceq 1507 ∈ wcel 2050 ∀wral 3082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-ral 3087 df-v 3411 |
This theorem is referenced by: rspcimedv 3531 rspcdv 3532 wrd2ind 13911 wrd2indOLD 13912 mreexd 16765 mreexexlemd 16767 catcocl 16808 catass 16809 moni 16858 subccocl 16967 funcco 16993 fullfo 17034 fthf1 17039 nati 17077 acsfiindd 17639 chpscmat 21148 friendshipgt3 27949 lmxrge0 30839 funressnfv 42683 |
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