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Mirrors > Home > ILE Home > Th. List > nfceqdf | GIF version |
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfceqdf.1 | ⊢ Ⅎ𝑥𝜑 |
nfceqdf.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
nfceqdf | ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfceqdf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | nfceqdf.2 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | eleq2d 2152 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | nfbidf 1473 | . . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵)) |
5 | 4 | albidv 1747 | . 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)) |
6 | df-nfc 2212 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
7 | df-nfc 2212 | . 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | |
8 | 5, 6, 7 | 3bitr4g 221 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1283 = wceq 1285 Ⅎwnf 1390 ∈ wcel 1434 Ⅎwnfc 2210 |
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 1377 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-4 1441 ax-17 1460 ax-ial 1468 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-cleq 2076 df-clel 2079 df-nfc 2212 |
This theorem is referenced by: nfopd 3613 dfnfc2 3645 nfimad 4736 nffvd 5260 |
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