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Mirrors > Home > ILE Home > Th. List > ee4anv | GIF version |
Description: Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.) |
Ref | Expression |
---|---|
ee4anv | ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 1643 | . . 3 ⊢ (∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓)) | |
2 | 1 | exbii 1585 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓)) |
3 | eeanv 1905 | . . 3 ⊢ (∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓)) | |
4 | 3 | 2exbii 1586 | . 2 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓)) |
5 | eeanv 1905 | . 2 ⊢ (∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) | |
6 | 2, 4, 5 | 3bitri 205 | 1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1469 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1438 |
This theorem is referenced by: ee8anv 1908 cgsex4g 2726 th3qlem1 6539 dmaddpq 7211 dmmulpq 7212 ltdcnq 7229 enq0ref 7265 nqpnq0nq 7285 nqnq0a 7286 nqnq0m 7287 genpdisj 7355 axaddcl 7696 axmulcl 7698 |
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