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Mirrors > Home > ILE Home > Th. List > opelopabsbALT | GIF version |
Description: The law of concretion in terms of substitutions. Less general than opelopabsb 3997, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
opelopabsbALT | ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 1554 | . . 3 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | vex 2560 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
3 | vex 2560 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
4 | 2, 3 | opth 3974 | . . . . . 6 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) |
5 | equcom 1593 | . . . . . . 7 ⊢ (𝑧 = 𝑥 ↔ 𝑥 = 𝑧) | |
6 | equcom 1593 | . . . . . . 7 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) | |
7 | 5, 6 | anbi12ci 434 | . . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑦 = 𝑤 ∧ 𝑥 = 𝑧)) |
8 | 4, 7 | bitri 173 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑦 = 𝑤 ∧ 𝑥 = 𝑧)) |
9 | 8 | anbi1i 431 | . . . 4 ⊢ ((〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
10 | 9 | 2exbii 1497 | . . 3 ⊢ (∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
11 | 1, 10 | bitri 173 | . 2 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
12 | elopab 3995 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
13 | 2sb5 1859 | . 2 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) | |
14 | 11, 12, 13 | 3bitr4i 201 | 1 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 [wsb 1645 〈cop 3378 {copab 3817 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 |
This theorem is referenced by: inopab 4468 cnvopab 4726 |
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