![]() |
Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > onfrALTlem4 | Structured version Visualization version GIF version |
Description: Lemma for onfrALT 40443. (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 3755 | . 2 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ∧ [𝑦 / 𝑥](𝑎 ∩ 𝑥) = ∅)) | |
2 | sbcel1v 3773 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎) | |
3 | vex 3443 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | sbceqg 4287 | . . . . 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 4312 | . . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) | |
7 | csbconstg 3834 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑎 = 𝑎) | |
8 | 3, 7 | ax-mp 5 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑎 = 𝑎 |
9 | csbvarg 4304 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦) | |
10 | 3, 9 | ax-mp 5 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 |
11 | 8, 10 | ineq12i 4113 | . . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝑎 ∩ ⦋𝑦 / 𝑥⦌𝑥) = (𝑎 ∩ 𝑦) |
12 | 6, 11 | eqtri 2821 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = (𝑎 ∩ 𝑦) |
13 | csb0 4285 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌∅ = ∅ | |
14 | 12, 13 | eqeq12i 2811 | . . . 4 ⊢ (⦋𝑦 / 𝑥⦌(𝑎 ∩ 𝑥) = ⦋𝑦 / 𝑥⦌∅ ↔ (𝑎 ∩ 𝑦) = ∅) |
15 | 5, 14 | bitri 276 | . . 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 276 | 1 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 [wsbc 3711 ⦋csb 3817 ∩ cin 3864 ∅c0 4217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-in 3872 df-nul 4218 |
This theorem is referenced by: onfrALTlem1 40442 onfrALTlem1VD 40784 |
Copyright terms: Public domain | W3C validator |