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Mirrors > Home > ILE Home > Th. List > ee8anv | GIF version |
Description: Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.) |
Ref | Expression |
---|---|
ee8anv | ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exrot4 1671 | . . 3 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓)) | |
2 | 1 | 2exbii 1586 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓)) |
3 | ee4anv 1914 | . . . 4 ⊢ (∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓)) | |
4 | 3 | 2exbii 1586 | . . 3 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓)) |
5 | 4 | 2exbii 1586 | . 2 ⊢ (∃𝑥∃𝑦∃𝑣∃𝑢∃𝑧∃𝑤∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓)) |
6 | ee4anv 1914 | . 2 ⊢ (∃𝑥∃𝑦∃𝑣∃𝑢(∃𝑧∃𝑤𝜑 ∧ ∃𝑡∃𝑠𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓)) | |
7 | 2, 5, 6 | 3bitri 205 | 1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1472 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-17 1506 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: enq0tr 7355 |
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