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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 148 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒)) |
4 | 1, 3 | rspcimdv 3450 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ¬ 𝜓 → ¬ 𝜒)) |
5 | 4 | con2d 129 | . 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓)) |
6 | dfrex2 3134 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓) | |
7 | 5, 6 | syl6ibr 242 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ∃wrex 3051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-v 3342 |
This theorem is referenced by: rspcedv 3453 scshwfzeqfzo 13772 symgfixfo 18059 slesolex 20690 usgr2pthlem 26869 clwlkclwwlkfo 27132 clwlksfoclwwlkOLD 27217 |
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