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Mirrors > Home > ILE Home > Th. List > nfbii | GIF version |
Description: Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
nfbii | ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | albii 1359 | . . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓) |
3 | 1, 2 | imbi12i 228 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)) |
4 | 3 | albii 1359 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) |
5 | df-nf 1350 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
6 | df-nf 1350 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) | |
7 | 4, 5, 6 | 3bitr4i 201 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 Ⅎwnf 1349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: nfxfr 1363 nfxfrd 1364 nfsb 1822 nfsbt 1850 hbsbd 1858 sbal1yz 1877 dvelimALT 1886 dvelimfv 1887 dvelimor 1894 nfeudv 1915 nfeuv 1918 nfceqi 2174 nfreudxy 2483 dfnfc2 3598 |
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