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Mirrors > Home > ILE Home > Th. List > cbvopab2v | GIF version |
Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.) |
Ref | Expression |
---|---|
cbvopab2v.1 | ⊢ (y = z → (φ ↔ ψ)) |
Ref | Expression |
---|---|
cbvopab2v | ⊢ {〈x, y〉 ∣ φ} = {〈x, z〉 ∣ ψ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 3541 | . . . . . . 7 ⊢ (y = z → 〈x, y〉 = 〈x, z〉) | |
2 | 1 | eqeq2d 2048 | . . . . . 6 ⊢ (y = z → (w = 〈x, y〉 ↔ w = 〈x, z〉)) |
3 | cbvopab2v.1 | . . . . . 6 ⊢ (y = z → (φ ↔ ψ)) | |
4 | 2, 3 | anbi12d 442 | . . . . 5 ⊢ (y = z → ((w = 〈x, y〉 ∧ φ) ↔ (w = 〈x, z〉 ∧ ψ))) |
5 | 4 | cbvexv 1792 | . . . 4 ⊢ (∃y(w = 〈x, y〉 ∧ φ) ↔ ∃z(w = 〈x, z〉 ∧ ψ)) |
6 | 5 | exbii 1493 | . . 3 ⊢ (∃x∃y(w = 〈x, y〉 ∧ φ) ↔ ∃x∃z(w = 〈x, z〉 ∧ ψ)) |
7 | 6 | abbii 2150 | . 2 ⊢ {w ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} = {w ∣ ∃x∃z(w = 〈x, z〉 ∧ ψ)} |
8 | df-opab 3810 | . 2 ⊢ {〈x, y〉 ∣ φ} = {w ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} | |
9 | df-opab 3810 | . 2 ⊢ {〈x, z〉 ∣ ψ} = {w ∣ ∃x∃z(w = 〈x, z〉 ∧ ψ)} | |
10 | 7, 8, 9 | 3eqtr4i 2067 | 1 ⊢ {〈x, y〉 ∣ φ} = {〈x, z〉 ∣ ψ} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 {cab 2023 〈cop 3370 {copab 3808 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 |
This theorem is referenced by: (None) |
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