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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 3056 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
2 | rspcimdv.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
4 | 3 | eleq1d 2815 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
5 | 4 | biimprd 251 | . . . . 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 3498 | . . 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 245 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2112 ∀wral 3051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 |
This theorem is referenced by: rspcimedv 3518 rspcdv 3519 wrd2ind 14253 mreexd 17099 mreexexlemd 17101 catcocl 17142 catass 17143 moni 17195 subccocl 17305 funcco 17331 fullfo 17373 fthf1 17378 nati 17416 acsfiindd 18013 chpscmat 21693 friendshipgt3 28435 lmxrge0 31570 funressnfv 44152 cfsetsnfsetf1 44168 |
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