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Mirrors > Home > MPE Home > Th. List > Mathboxes > onfrALTlem4 | Structured version Visualization version GIF version |
Description: Lemma for onfrALT 43612. (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 3828 | . 2 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅)) | |
2 | sbcel1v 3847 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎) | |
3 | vex 3476 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | sbceqg 4408 | . . . . 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 4438 | . . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) | |
7 | csbconstg 3911 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑎 = 𝑎) | |
8 | 3, 7 | ax-mp 5 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎 |
9 | csbvarg 4430 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦) | |
10 | 3, 9 | ax-mp 5 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 |
11 | 8, 10 | ineq12i 4209 | . . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = (𝑎 ∩ 𝑦) |
12 | 6, 11 | eqtri 2758 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦) |
13 | csb0 4406 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅ | |
14 | 12, 13 | eqeq12i 2748 | . . . 4 ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅ ↔ (𝑎 ∩ 𝑦) = ∅) |
15 | 5, 14 | bitri 274 | . . 3 ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎 ∩ 𝑦) = ∅) |
16 | 2, 15 | anbi12i 625 | . 2 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
17 | 1, 16 | bitri 274 | 1 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 [wsbc 3776 ⦋csb 3892 ∩ cin 3946 ∅c0 4321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-in 3954 df-nul 4322 |
This theorem is referenced by: onfrALTlem1 43611 onfrALTlem1VD 43953 |
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