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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 3140 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
2 | rspcimdv.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
4 | 3 | eleq1d 2894 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
5 | 4 | biimprd 249 | . . . . 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 3589 | . . 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 243 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 df-ral 3140 |
This theorem is referenced by: rspcimedv 3611 rspcdv 3612 wrd2ind 14073 mreexd 16901 mreexexlemd 16903 catcocl 16944 catass 16945 moni 16994 subccocl 17103 funcco 17129 fullfo 17170 fthf1 17175 nati 17213 acsfiindd 17775 chpscmat 21378 friendshipgt3 28104 lmxrge0 31094 funressnfv 43155 |
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