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Mirrors > Home > ILE Home > Th. List > rexdifpr | GIF version |
Description: Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.) |
Ref | Expression |
---|---|
rexdifpr | ⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifpr 3583 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) | |
2 | 3anass 967 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶))) | |
3 | 1, 2 | bitri 183 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶))) |
4 | 3 | anbi1i 454 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) ∧ 𝜑)) |
5 | anass 399 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ∧ 𝜑))) | |
6 | df-3an 965 | . . . . . 6 ⊢ ((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑) ↔ ((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ∧ 𝜑)) | |
7 | 6 | bicomi 131 | . . . . 5 ⊢ (((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ∧ 𝜑) ↔ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑)) |
8 | 7 | anbi2i 453 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))) |
9 | 5, 8 | bitri 183 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))) |
10 | 4, 9 | bitri 183 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))) |
11 | 10 | rexbii2 2465 | 1 ⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 963 ∈ wcel 2125 ≠ wne 2324 ∃wrex 2433 ∖ cdif 3095 {cpr 3557 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-rex 2438 df-v 2711 df-dif 3100 df-un 3102 df-sn 3562 df-pr 3563 |
This theorem is referenced by: (None) |
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