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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 2247 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | nfbidf 1539 | . . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵)) |
5 | 4 | albidv 1824 | . 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)) |
6 | df-nfc 2308 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
7 | df-nfc 2308 | . 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | |
8 | 5, 6, 7 | 3bitr4g 223 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 = wceq 1353 Ⅎwnf 1460 ∈ wcel 2148 Ⅎwnfc 2306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-cleq 2170 df-clel 2173 df-nfc 2308 |
This theorem is referenced by: nfopd 3797 dfnfc2 3829 nfimad 4981 nffvd 5529 |
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