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Mirrors > Home > MPE Home > Th. List > rexdifsn | Structured version Visualization version GIF version |
Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.) |
Ref | Expression |
---|---|
rexdifsn | ⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4260 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵)) | |
2 | 1 | anbi1i 727 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) ∧ 𝜑)) |
3 | anass 679 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝜑))) | |
4 | 2, 3 | bitri 263 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝜑))) |
5 | 4 | rexbii2 3021 | 1 ⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∖ cdif 3537 {csn 4125 |
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 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rex 2902 df-v 3175 df-dif 3543 df-sn 4126 |
This theorem is referenced by: symgfix2 17659 2spot2iun2spont 26418 dihatexv 35645 lcfl8b 35811 usgr2pth0 40971 wspniunwspnon 41130 |
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