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| Mirrors > Home > MPE Home > Th. List > spcimedv | Structured version Visualization version GIF version | ||
| Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| spcimdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) | 
| spcimedv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) | 
| Ref | Expression | 
|---|---|
| spcimedv | ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | spcimdv.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
| 2 | spcimedv.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) | |
| 3 | 2 | con3d 152 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒)) | 
| 4 | 1, 3 | spcimdv 3592 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒)) | 
| 5 | 4 | con2d 134 | . 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓)) | 
| 6 | df-ex 1779 | . 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓) | |
| 7 | 5, 6 | imbitrrdi 252 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1537 = wceq 1539 ∃wex 1778 ∈ wcel 2107 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-clel 2815 | 
| This theorem is referenced by: spc3egv 3602 hashf1rn 14392 cshwsexaOLD 14864 wwlktovfo 14998 uvcendim 21868 wlkiswwlks2 29896 wwlksnextsurj 29921 elwwlks2 29987 elwspths2spth 29988 clwlkclwwlklem1 30019 sticksstones4 42151 rtrclex 43635 clcnvlem 43641 iunrelexpuztr 43737 | 
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