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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 1457 | . . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓) |
3 | 1, 2 | imbi12i 238 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)) |
4 | 3 | albii 1457 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) |
5 | df-nf 1448 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
6 | df-nf 1448 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) | |
7 | 4, 5, 6 | 3bitr4i 211 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 Ⅎwnf 1447 |
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 1434 ax-gen 1436 |
This theorem depends on definitions: df-bi 116 df-nf 1448 |
This theorem is referenced by: nfxfr 1461 nfxfrd 1462 nfsb 1933 nfsbt 1963 hbsbd 1969 sbal1yz 1988 dvelimALT 1997 dvelimfv 1998 dvelimor 2005 nfeudv 2028 nfeuv 2031 nfceqi 2302 nfreudxy 2637 dfnfc2 3801 |
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