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Mirrors > Home > MPE Home > Th. List > cbvopab1s | Structured version Visualization version GIF version |
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.) |
Ref | Expression |
---|---|
cbvopab1s | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
2 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 〈𝑧, 𝑦〉 | |
3 | nfs1v 2425 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
4 | 2, 3 | nfan 1816 | . . . . 5 ⊢ Ⅎ𝑥(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
5 | 4 | nfex 2140 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
6 | opeq1 4340 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
7 | 6 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑧, 𝑦〉)) |
8 | sbequ12 2097 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
9 | 7, 8 | anbi12d 743 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
10 | 9 | exbidv 1837 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
11 | 1, 5, 10 | cbvex 2260 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)) |
12 | 11 | abbii 2726 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} |
13 | df-opab 4644 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
14 | df-opab 4644 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} | |
15 | 12, 13, 14 | 3eqtr4i 2642 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 [wsb 1867 {cab 2596 〈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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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: (None) |
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