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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 3540 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒)) |
5 | 4 | con2d 136 | . 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓)) |
6 | df-ex 1782 | . 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓) | |
7 | 5, 6 | syl6ibr 255 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: spc3egv 3552 hashf1rn 13709 cshwsexa 14177 wwlktovfo 14313 uvcendim 20536 wlkiswwlks2 27661 wwlksnextsurj 27686 elwwlks2 27752 elwspths2spth 27753 clwlkclwwlklem1 27784 rtrclex 40317 clcnvlem 40323 iunrelexpuztr 40420 |
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