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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 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
2 | reean.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | 1, 2 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
4 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 | |
5 | reean.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
6 | 4, 5 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ 𝜓) |
7 | 3, 6 | eean 2368 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))) |
8 | 7 | reeanlem 3368 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 Ⅎwnf 1783 ∈ wcel 2113 ∃wrex 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-ral 3146 df-rex 3147 |
This theorem is referenced by: disjrnmpt2 41455 |
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