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Mirrors > Home > ILE Home > Th. List > exancom | GIF version |
Description: Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
exancom | ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 264 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
2 | 1 | exbii 1567 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1451 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1406 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-4 1470 ax-ial 1497 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.29r 1583 19.42h 1648 19.42 1649 risset 2437 morex 2837 dfuni2 3704 eluni2 3706 unipr 3716 dfiun2g 3811 uniuni 4332 cnvco 4684 imadif 5161 funimaexglem 5164 bdcuni 12766 bj-axun2 12805 |
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