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Mirrors > Home > MPE Home > Th. List > reean | Structured version Visualization version GIF version |
Description: Rearrange restricted existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
reean.1 | ⊢ Ⅎ𝑦𝜑 |
reean.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
reean | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an4 861 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝜑 ∧ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) | |
2 | 1 | 2exbii 1765 | . . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝜑 ∧ 𝜓)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) |
3 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
4 | reean.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
5 | 3, 4 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
6 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 | |
7 | reean.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
8 | 6, 7 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ 𝜓) |
9 | 5, 8 | eean 2169 | . . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))) |
10 | 2, 9 | bitri 263 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))) |
11 | r2ex 3043 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝜑 ∧ 𝜓))) | |
12 | df-rex 2902 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
13 | df-rex 2902 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓)) | |
14 | 12, 13 | anbi12i 729 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))) |
15 | 10, 11, 14 | 3bitr4i 291 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-ral 2901 df-rex 2902 |
This theorem is referenced by: reeanv 3086 disjrnmpt2 38370 |
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