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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 155 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒)) |
4 | 1, 3 | spcimdv 3589 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒)) |
5 | 4 | con2d 136 | . 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓)) |
6 | df-ex 1772 | . 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓) | |
7 | 5, 6 | syl6ibr 253 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 |
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 |
This theorem is referenced by: spc3egv 3601 hashf1rn 13701 cshwsexa 14174 wwlktovfo 14310 uvcendim 20919 wlkiswwlks2 27580 wwlksnextsurj 27605 elwwlks2 27672 elwspths2spth 27673 clwlkclwwlklem1 27704 rtrclex 39855 clcnvlem 39861 iunrelexpuztr 39942 |
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