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Mirrors > Home > ILE Home > Th. List > exlimdd | GIF version |
Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
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.3 | . 2 ⊢ (𝜑 → ∃𝑥𝜓) | |
2 | exlimdd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | exlimdd.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
4 | exlimdd.4 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
5 | 4 | ex 115 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
6 | 2, 3, 5 | exlimd 1607 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
7 | 1, 6 | mpd 13 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 Ⅎwnf 1470 ∃wex 1502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia3 108 ax-5 1457 ax-gen 1459 ax-ie2 1504 ax-4 1520 |
This theorem depends on definitions: df-bi 117 df-nf 1471 |
This theorem is referenced by: fvmptdf 5616 ovmpodf 6019 exmidfodomrlemr 7214 exmidfodomrlemrALT 7215 ltexprlemm 7612 dfgrp3mlem 12992 |
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