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Mirrors > Home > MPE Home > Th. List > 3reeanv | Structured version Visualization version GIF version |
Description: Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.) |
Ref | Expression |
---|---|
3reeanv | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.41v 3184 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒)) | |
2 | reeanv 3222 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) | |
3 | 1, 2 | bianbi 625 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒)) |
4 | df-3an 1086 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
5 | 4 | 2rexbii 3125 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
6 | reeanv 3222 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒)) | |
7 | 5, 6 | bitri 274 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒)) |
8 | 7 | rexbii 3090 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒)) |
9 | df-3an 1086 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒)) | |
10 | 3, 8, 9 | 3bitr4i 302 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∃wrex 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-ex 1774 df-ral 3058 df-rex 3067 |
This theorem is referenced by: poxp2 8152 poxp3 8159 imasmnd2 18736 imasgrp2 19016 imasrng 20122 imasring 20271 axeuclid 28792 lshpkrlem6 38591 |
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