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Mirrors > Home > MPE Home > Th. List > exlimdd | Structured version Visualization version GIF version |
Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 3-Sep-2023.) |
Ref | Expression |
---|---|
exlimdd.1 | ⊢ Ⅎ𝑥𝜑 |
exlimdd.2 | ⊢ Ⅎ𝑥𝜒 |
exlimdd.3 | ⊢ (𝜑 → ∃𝑥𝜓) |
exlimdd.4 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
exlimdd | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimdd.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | exlimdd.2 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | exlimdd.3 | . 2 ⊢ (𝜑 → ∃𝑥𝜓) | |
4 | exlimdd.4 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
5 | 4 | ex 416 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) |
6 | 1, 2, 3, 5 | exlimimdd 2219 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1787 Ⅎwnf 1791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-12 2177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 |
This theorem is referenced by: fvmptd3f 6855 ovmpodf 7387 ex-natded9.26 28534 stoweidlem43 43305 stoweidlem44 43306 stoweidlem54 43316 |
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