![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rexex | GIF version |
Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.) |
Ref | Expression |
---|---|
rexex | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2461 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | simpr 110 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑) | |
3 | 2 | eximi 1600 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥𝜑) |
4 | 1, 3 | sylbi 121 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-rex 2461 |
This theorem is referenced by: reu3 2927 rmo2i 3053 dffo5 5660 halfnq 7388 nsmallnq 7390 0npr 7460 genpml 7494 genpmu 7495 ltexprlemm 7577 ltexprlemloc 7584 dedekindeulemlub 13731 dedekindicclemlub 13740 |
Copyright terms: Public domain | W3C validator |