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Mirrors > Home > MPE Home > Th. List > nfbidf | Structured version Visualization version GIF version |
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
Ref | Expression |
---|---|
albid.1 | ⊢ Ⅎ𝑥𝜑 |
albid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
nfbidf | ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | albid.2 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | exbid 2223 | . . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
4 | 1, 2 | albid 2222 | . . 3 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
5 | 3, 4 | imbi12d 348 | . 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒))) |
6 | df-nf 1786 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓)) | |
7 | df-nf 1786 | . 2 ⊢ (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒)) | |
8 | 5, 6, 7 | 3bitr4g 317 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∃wex 1781 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1782 df-nf 1786 |
This theorem is referenced by: drnf2 2455 dvelimdf 2460 nfceqdf 2951 nfabdw 2976 wl-nfimf1 34931 ichnfimlem 43980 |
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