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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 5400 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 2741 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉) | |
2 | vex 3405 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3405 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth 5349 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) |
5 | 1, 4 | bitri 278 | . . . 4 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) |
6 | 5 | anbi1i 627 | . . 3 ⊢ ((〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) |
7 | 6 | 2exbii 1856 | . 2 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) |
8 | elopab 5397 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
9 | 2sb5 2276 | . 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) | |
10 | 7, 8, 9 | 3bitr4i 306 | 1 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 [wsb 2070 ∈ wcel 2110 〈cop 4537 {copab 5105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-opab 5106 |
This theorem is referenced by: opelopabsb 5400 inopab 5688 cnvopab 5991 brabsb2 36570 |
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