![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cleqf | GIF version |
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2254. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
cleqf.1 | ⊢ Ⅎ𝑥𝐴 |
cleqf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
cleqf | ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2148 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | nfv 1505 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) | |
3 | cleqf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfcri 2290 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
5 | cleqf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfcri 2290 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
7 | 4, 6 | nfbi 1566 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
8 | eleq1 2217 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
9 | eleq1 2217 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
10 | 8, 9 | bibi12d 234 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))) |
11 | 2, 7, 10 | cbval 1731 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
12 | 1, 11 | bitr4i 186 | 1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1330 = wceq 1332 ∈ wcel 2125 Ⅎwnfc 2283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1740 df-cleq 2147 df-clel 2150 df-nfc 2285 |
This theorem is referenced by: abid2f 2322 n0rf 3402 eq0 3408 iunab 3891 iinab 3906 sniota 5155 |
Copyright terms: Public domain | W3C validator |