![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfbidv | Structured version Visualization version GIF version |
Description: An equality theorem for nonfreeness. See nfbidf 2209 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1963, ax-7 2003, ax-12 2163 by adapting proof of nfbidf 2209. (Revised by BJ, 25-Sep-2022.) |
Ref | Expression |
---|---|
albidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
nfbidv | ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albidv.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | exbidv 1916 | . . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
3 | 1 | albidv 1915 | . . 3 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
4 | 2, 3 | imbi12d 344 | . 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒))) |
5 | df-nf 1778 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓)) | |
6 | df-nf 1778 | . 2 ⊢ (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒)) | |
7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 |
This theorem depends on definitions: df-bi 206 df-ex 1774 df-nf 1778 |
This theorem is referenced by: nfcjust 2876 nfcr 2880 nfcriOLD 2885 nfcriOLDOLD 2886 bj-drnf2v 36189 |
Copyright terms: Public domain | W3C validator |