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Mirrors > Home > ILE Home > Th. List > nfcjust | GIF version |
Description: Justification theorem for df-nfc 2285. (Contributed by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
nfcjust | ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1505 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
2 | eleq1 2217 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
3 | 1, 2 | nfbidf 1516 | . 2 ⊢ (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑧 ∈ 𝐴)) |
4 | 3 | cbvalv 1894 | 1 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1330 Ⅎwnf 1437 ∈ wcel 2125 |
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-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-cleq 2147 df-clel 2150 |
This theorem is referenced by: (None) |
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