Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vopelopabsb | Structured version Visualization version GIF version |
Description: The law of concretion in terms of substitutions. Version of opelopabsb 5443 with set variables. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Remove unnecessary commutation. (Revised by SN, 1-Sep-2024.) |
Ref | Expression |
---|---|
vopelopabsb | ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2745 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉) | |
2 | vex 3436 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3436 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth 5391 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) |
5 | 1, 4 | bitri 274 | . . . 4 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) |
6 | 5 | anbi1i 624 | . . 3 ⊢ ((〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) |
7 | 6 | 2exbii 1851 | . 2 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) |
8 | elopab 5440 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
9 | 2sb5 2272 | . 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) | |
10 | 7, 8, 9 | 3bitr4i 303 | 1 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 [wsb 2067 ∈ wcel 2106 〈cop 4567 {copab 5136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-opab 5137 |
This theorem is referenced by: opelopabsb 5443 inopab 5739 cnvopab 6042 brabsb2 36876 |
Copyright terms: Public domain | W3C validator |