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Mirrors > Home > ILE Home > Th. List > nfeq | GIF version |
Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfnfc.1 | ⊢ Ⅎ𝑥𝐴 |
nfeq.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfeq | ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2075 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) | |
2 | nfnfc.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2213 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
4 | nfeq.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2213 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
6 | 3, 5 | nfbi 1521 | . . 3 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) |
7 | 6 | nfal 1508 | . 2 ⊢ Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵) |
8 | 1, 7 | nfxfr 1403 | 1 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∀wal 1282 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 Ⅎwnfc 2206 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-cleq 2074 df-clel 2077 df-nfc 2208 |
This theorem is referenced by: nfel 2227 nfeq1 2228 nfeq2 2230 nfne 2337 raleqf 2545 rexeqf 2546 reueq1f 2547 rmoeq1f 2548 rabeqf 2594 sbceqg 2922 csbhypf 2941 nfiotadxy 4890 nffn 5015 nffo 5125 fvmptdf 5279 mpteqb 5282 fvmptf 5284 eqfnfv2f 5290 dff13f 5430 ovmpt2s 5644 ov2gf 5645 ovmpt2dxf 5646 ovmpt2df 5652 eqerlem 6160 |
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