Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > exsimpl | Structured version Visualization version 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 483 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | eximi 1826 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 |
This theorem is referenced by: 19.40 1878 moexexlem 2704 elisset 3503 elex 3510 sbc5 3797 r19.2zb 4437 dmcoss 5835 suppimacnvss 7829 unblem2 8759 kmlem8 9571 isssc 17078 bnj1143 31961 bnj1371 32198 bnj1374 32200 bj-elissetv 34088 atex 36422 rtrclex 39855 clcnvlem 39861 pm10.55 40578 |
Copyright terms: Public domain | W3C validator |