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Mirrors > Home > MPE Home > Th. List > Mathboxes > onfrALTlem4 | Structured version Visualization version GIF version |
Description: Lemma for onfrALT 43613. (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 3829 | . 2 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅)) | |
2 | sbcel1v 3848 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎) | |
3 | vex 3477 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | sbceqg 4409 | . . . . 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 4439 | . . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) | |
7 | csbconstg 3912 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑎 = 𝑎) | |
8 | 3, 7 | ax-mp 5 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎 |
9 | csbvarg 4431 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦) | |
10 | 3, 9 | ax-mp 5 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 |
11 | 8, 10 | ineq12i 4210 | . . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = (𝑎 ∩ 𝑦) |
12 | 6, 11 | eqtri 2759 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦) |
13 | csb0 4407 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅ | |
14 | 12, 13 | eqeq12i 2749 | . . . 4 ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅ ↔ (𝑎 ∩ 𝑦) = ∅) |
15 | 5, 14 | bitri 275 | . . 3 ⊢ ([𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅ ↔ (𝑎 ∩ 𝑦) = ∅) |
16 | 2, 15 | anbi12i 626 | . 2 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
17 | 1, 16 | bitri 275 | 1 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 [wsbc 3777 ⦋csb 3893 ∩ cin 3947 ∅c0 4322 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-in 3955 df-nul 4323 |
This theorem is referenced by: onfrALTlem1 43612 onfrALTlem1VD 43954 |
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