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Mirrors > Home > MPE Home > Th. List > rexdifpr | Structured version Visualization version 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 4557 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) | |
2 | 3anass 1092 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶))) | |
3 | 1, 2 | bitri 278 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶))) |
4 | 3 | anbi1i 626 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) ∧ 𝜑)) |
5 | anass 472 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ∧ 𝜑))) | |
6 | df-3an 1086 | . . . . . 6 ⊢ ((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑) ↔ ((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ∧ 𝜑)) | |
7 | 6 | bicomi 227 | . . . . 5 ⊢ (((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ∧ 𝜑) ↔ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑)) |
8 | 7 | anbi2i 625 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))) |
9 | 5, 8 | bitri 278 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))) |
10 | 4, 9 | bitri 278 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))) |
11 | 10 | rexbii2 3208 | 1 ⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ∖ cdif 3878 {cpr 4527 |
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-or 845 df-3an 1086 df-tru 1541 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-un 3886 df-sn 4526 df-pr 4528 |
This theorem is referenced by: usgr2pth0 27554 |
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