Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cbvopab2 | Structured version Visualization version GIF version |
Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.) |
Ref | Expression |
---|---|
cbvopab2.1 | ⊢ Ⅎ𝑧𝜑 |
cbvopab2.2 | ⊢ Ⅎ𝑦𝜓 |
cbvopab2.3 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopab2 | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉 | |
2 | cbvopab2.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfan 1816 | . . . . 5 ⊢ Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤 = 〈𝑥, 𝑧〉 | |
5 | cbvopab2.2 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
6 | 4, 5 | nfan 1816 | . . . . 5 ⊢ Ⅎ𝑦(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓) |
7 | opeq2 4341 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑧〉) | |
8 | 7 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑥, 𝑧〉)) |
9 | cbvopab2.3 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | anbi12d 743 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓))) |
11 | 3, 6, 10 | cbvex 2260 | . . . 4 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)) |
12 | 11 | exbii 1764 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)) |
13 | 12 | abbii 2726 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)} |
14 | df-opab 4644 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
15 | df-opab 4644 | . 2 ⊢ {〈𝑥, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)} | |
16 | 13, 14, 15 | 3eqtr4i 2642 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 {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: cbvoprab3 6629 |
Copyright terms: Public domain | W3C validator |