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Mirrors > Home > ILE Home > Th. List > eeor | GIF version |
Description: Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.) |
Ref | Expression |
---|---|
eeor.1 | ⊢ Ⅎ𝑦𝜑 |
eeor.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
eeor | ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eeor.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | 19.45 1613 | . . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓)) |
3 | 2 | exbii 1536 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓)) |
4 | eeor.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfex 1568 | . . 3 ⊢ Ⅎ𝑥∃𝑦𝜓 |
6 | 5 | 19.44 1612 | . 2 ⊢ (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) |
7 | 3, 6 | bitri 182 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∨ wo 661 Ⅎwnf 1389 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: (None) |
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