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| Mirrors > Home > MPE Home > Th. List > rspcimedv | Structured version Visualization version GIF version | ||
| Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| rspcimdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) | 
| rspcimedv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) | 
| Ref | Expression | 
|---|---|
| rspcimedv | ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rspcimdv.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
| 2 | rspcimedv.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) | |
| 3 | 2 | con3d 152 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒)) | 
| 4 | 1, 3 | rspcimdv 3612 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ¬ 𝜓 → ¬ 𝜒)) | 
| 5 | 4 | con2d 134 | . 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓)) | 
| 6 | dfrex2 3073 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓) | |
| 7 | 5, 6 | imbitrrdi 252 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 | 
| This theorem is referenced by: rspcedv 3615 scshwfzeqfzo 14865 symgfixfo 19457 slesolex 22688 usgr2pthlem 29783 clwlkclwwlkfo 30028 satfdmlem 35373 | 
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