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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 4190, 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 1643 | . . 3 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | vex 2692 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
3 | vex 2692 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
4 | 2, 3 | opth 4167 | . . . . . 6 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) |
5 | equcom 1683 | . . . . . . 7 ⊢ (𝑧 = 𝑥 ↔ 𝑥 = 𝑧) | |
6 | equcom 1683 | . . . . . . 7 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) | |
7 | 5, 6 | anbi12ci 457 | . . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑦 = 𝑤 ∧ 𝑥 = 𝑧)) |
8 | 4, 7 | bitri 183 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑦 = 𝑤 ∧ 𝑥 = 𝑧)) |
9 | 8 | anbi1i 454 | . . . 4 ⊢ ((〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
10 | 9 | 2exbii 1586 | . . 3 ⊢ (∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
11 | 1, 10 | bitri 183 | . 2 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
12 | elopab 4188 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
13 | 2sb5 1959 | . 2 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) | |
14 | 11, 12, 13 | 3bitr4i 211 | 1 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1332 ∃wex 1469 ∈ wcel 1481 [wsb 1736 〈cop 3535 {copab 3996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 |
This theorem is referenced by: inopab 4679 cnvopab 4948 |
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