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Mirrors > Home > MPE Home > Th. List > Mathboxes > onfrALTlem4 | Structured version Visualization version GIF version |
Description: Lemma for onfrALT 40890. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
onfrALTlem4 | ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcan 3823 | . 2 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅)) | |
2 | sbcel1v 3841 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎) | |
3 | vex 3499 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | sbceqg 4363 | . . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅) |
6 | csbin 4393 | . . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) | |
7 | csbconstg 3904 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑎 = 𝑎) | |
8 | 3, 7 | ax-mp 5 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎 |
9 | csbvarg 4385 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦) | |
10 | 3, 9 | ax-mp 5 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 |
11 | 8, 10 | ineq12i 4189 | . . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = (𝑎 ∩ 𝑦) |
12 | 6, 11 | eqtri 2846 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦) |
13 | csb0 4361 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅ | |
14 | 12, 13 | eqeq12i 2838 | . . . 4 ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅ ↔ (𝑎 ∩ 𝑦) = ∅) |
15 | 5, 14 | bitri 277 | . . 3 ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎 ∩ 𝑦) = ∅) |
16 | 2, 15 | anbi12i 628 | . 2 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
17 | 1, 16 | bitri 277 | 1 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 [wsbc 3774 ⦋csb 3885 ∩ cin 3937 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-in 3945 df-nul 4294 |
This theorem is referenced by: onfrALTlem1 40889 onfrALTlem1VD 41231 |
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