Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 1399 | . . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓) |
3 | 1, 2 | imbi12i 237 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)) |
4 | 3 | albii 1399 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) |
5 | df-nf 1390 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
6 | df-nf 1390 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) | |
7 | 4, 5, 6 | 3bitr4i 210 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 |
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-gen 1378 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: nfxfr 1403 nfxfrd 1404 nfsb 1863 nfsbt 1891 hbsbd 1899 sbal1yz 1918 dvelimALT 1927 dvelimfv 1928 dvelimor 1935 nfeudv 1956 nfeuv 1959 nfceqi 2215 nfreudxy 2527 dfnfc2 3619 |
Copyright terms: Public domain | W3C validator |