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Mirrors > Home > ILE Home > Th. List > rexcom13 | GIF version |
Description: Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.) |
Ref | Expression |
---|---|
rexcom13 | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom 2553 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑) | |
2 | rexcom 2553 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑) | |
3 | 2 | rexbii 2401 | . 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑) |
4 | rexcom 2553 | . 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) | |
5 | 1, 3, 4 | 3bitri 205 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wrex 2376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rex 2381 |
This theorem is referenced by: rexrot4 2555 |
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