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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 1592 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.29r 1608 19.42h 1674 19.42 1675 risset 2492 morex 2905 dfuni2 3785 eluni2 3787 unipr 3797 dfiun2g 3892 uniuni 4423 cnvco 4783 imadif 5262 funimaexglem 5265 pceu 12206 bdcuni 13599 bj-axun2 13638 |
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