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Mirrors > Home > ILE Home > Th. List > nfcjust | GIF version |
Description: Justification theorem for df-nfc 2208. (Contributed by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
nfcjust | ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1461 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
2 | eleq1 2141 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
3 | 1, 2 | nfbidf 1472 | . 2 ⊢ (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑧 ∈ 𝐴)) |
4 | 3 | cbvalv 1835 | 1 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 ∈ wcel 1433 |
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-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-cleq 2074 df-clel 2077 |
This theorem is referenced by: (None) |
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