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Mirrors > Home > ILE Home > Th. List > exsimpl | GIF version |
Description: Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
exsimpl | ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | eximi 1600 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃wex 1492 |
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 |
This theorem is referenced by: 19.40 1631 euex 2056 moexexdc 2110 elex 2750 sbc5 2988 dmcoss 4898 fmptco 5684 brabvv 5923 brtpos2 6254 |
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