![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > caovcang | GIF version |
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovcang.1 | ⊢ ((φ ∧ (x ∈ 𝑇 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z)) |
Ref | Expression |
---|---|
caovcang | ⊢ ((φ ∧ (A ∈ 𝑇 ∧ B ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovcang.1 | . . 3 ⊢ ((φ ∧ (x ∈ 𝑇 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z)) | |
2 | 1 | ralrimivvva 2396 | . 2 ⊢ (φ → ∀x ∈ 𝑇 ∀y ∈ 𝑆 ∀z ∈ 𝑆 ((x𝐹y) = (x𝐹z) ↔ y = z)) |
3 | oveq1 5462 | . . . . 5 ⊢ (x = A → (x𝐹y) = (A𝐹y)) | |
4 | oveq1 5462 | . . . . 5 ⊢ (x = A → (x𝐹z) = (A𝐹z)) | |
5 | 3, 4 | eqeq12d 2051 | . . . 4 ⊢ (x = A → ((x𝐹y) = (x𝐹z) ↔ (A𝐹y) = (A𝐹z))) |
6 | 5 | bibi1d 222 | . . 3 ⊢ (x = A → (((x𝐹y) = (x𝐹z) ↔ y = z) ↔ ((A𝐹y) = (A𝐹z) ↔ y = z))) |
7 | oveq2 5463 | . . . . 5 ⊢ (y = B → (A𝐹y) = (A𝐹B)) | |
8 | 7 | eqeq1d 2045 | . . . 4 ⊢ (y = B → ((A𝐹y) = (A𝐹z) ↔ (A𝐹B) = (A𝐹z))) |
9 | eqeq1 2043 | . . . 4 ⊢ (y = B → (y = z ↔ B = z)) | |
10 | 8, 9 | bibi12d 224 | . . 3 ⊢ (y = B → (((A𝐹y) = (A𝐹z) ↔ y = z) ↔ ((A𝐹B) = (A𝐹z) ↔ B = z))) |
11 | oveq2 5463 | . . . . 5 ⊢ (z = 𝐶 → (A𝐹z) = (A𝐹𝐶)) | |
12 | 11 | eqeq2d 2048 | . . . 4 ⊢ (z = 𝐶 → ((A𝐹B) = (A𝐹z) ↔ (A𝐹B) = (A𝐹𝐶))) |
13 | eqeq2 2046 | . . . 4 ⊢ (z = 𝐶 → (B = z ↔ B = 𝐶)) | |
14 | 12, 13 | bibi12d 224 | . . 3 ⊢ (z = 𝐶 → (((A𝐹B) = (A𝐹z) ↔ B = z) ↔ ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))) |
15 | 6, 10, 14 | rspc3v 2659 | . 2 ⊢ ((A ∈ 𝑇 ∧ B ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀x ∈ 𝑇 ∀y ∈ 𝑆 ∀z ∈ 𝑆 ((x𝐹y) = (x𝐹z) ↔ y = z) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))) |
16 | 2, 15 | mpan9 265 | 1 ⊢ ((φ ∧ (A ∈ 𝑇 ∧ B ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 ∀wral 2300 (class class class)co 5455 |
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-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-iota 4810 df-fv 4853 df-ov 5458 |
This theorem is referenced by: caovcand 5605 |
Copyright terms: Public domain | W3C validator |