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Mirrors > Home > MPE Home > Th. List > opelopabsbALT | Structured version Visualization version GIF version |
Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4910, 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 2029 | . . 3 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | vex 3176 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
3 | vex 3176 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
4 | 2, 3 | opth 4871 | . . . . . 6 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) |
5 | equcom 1932 | . . . . . . 7 ⊢ (𝑧 = 𝑥 ↔ 𝑥 = 𝑧) | |
6 | equcom 1932 | . . . . . . 7 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) | |
7 | 5, 6 | anbi12ci 730 | . . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑦 = 𝑤 ∧ 𝑥 = 𝑧)) |
8 | 4, 7 | bitri 263 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑦 = 𝑤 ∧ 𝑥 = 𝑧)) |
9 | 8 | anbi1i 727 | . . . 4 ⊢ ((〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
10 | 9 | 2exbii 1765 | . . 3 ⊢ (∃𝑦∃𝑥(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
11 | 1, 10 | bitri 263 | . 2 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) |
12 | elopab 4908 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
13 | 2sb5 2431 | . 2 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑)) | |
14 | 11, 12, 13 | 3bitr4i 291 | 1 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 [wsb 1867 ∈ wcel 1977 〈cop 4131 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 |
This theorem is referenced by: inopab 5174 cnvopab 5452 brabsb2 33165 |
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