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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 3550 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ¬ 𝜓 → ¬ 𝜒)) |
5 | 4 | con2d 134 | . 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓)) |
6 | dfrex2 3169 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓) | |
7 | 5, 6 | syl6ibr 251 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 |
This theorem is referenced by: rspcedv 3553 scshwfzeqfzo 14537 symgfixfo 19045 slesolex 21829 usgr2pthlem 28128 clwlkclwwlkfo 28370 satfdmlem 33327 |
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