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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 3595 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒)) |
5 | 4 | con2d 136 | . 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓)) |
6 | df-ex 1780 | . 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓) | |
7 | 5, 6 | syl6ibr 254 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1534 = wceq 1536 ∃wex 1779 ∈ wcel 2113 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-cleq 2817 df-clel 2896 |
This theorem is referenced by: spc3egv 3607 hashf1rn 13716 cshwsexa 14189 wwlktovfo 14325 uvcendim 20994 wlkiswwlks2 27656 wwlksnextsurj 27681 elwwlks2 27748 elwspths2spth 27749 clwlkclwwlklem1 27780 rtrclex 39983 clcnvlem 39989 iunrelexpuztr 40070 |
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