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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 4680 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵)) | |
2 | 1 | anbi1i 626 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) ∧ 𝜑)) |
3 | anass 472 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝜑))) | |
4 | 2, 3 | bitri 278 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝜑))) |
5 | 4 | rexbii2 3208 | 1 ⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ∖ cdif 3878 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-rex 3112 df-v 3443 df-dif 3884 df-sn 4526 |
This theorem is referenced by: symgfix2 18536 usgr2pth0 27554 wspniunwspnon 27709 dihatexv 38634 lcfl8b 38800 |
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