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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 3554 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒)) |
5 | 4 | con2d 134 | . 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓)) |
6 | df-ex 1783 | . 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓) | |
7 | 5, 6 | syl6ibr 252 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-clel 2811 |
This theorem is referenced by: spc3egv 3564 hashf1rn 14261 cshwsexaOLD 14722 wwlktovfo 14856 uvcendim 21276 wlkiswwlks2 28869 wwlksnextsurj 28894 elwwlks2 28960 elwspths2spth 28961 clwlkclwwlklem1 28992 sticksstones4 40607 rtrclex 41981 clcnvlem 41987 iunrelexpuztr 42083 |
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