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Mirrors > Home > MPE Home > Th. List > nfbidv | Structured version Visualization version GIF version |
Description: An equality theorem for nonfreeness. See nfbidf 2224 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1976, ax-7 2018, ax-12 2177 by adapting proof of nfbidf 2224. (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 1929 | . . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
3 | 1 | albidv 1928 | . . 3 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
4 | 2, 3 | imbi12d 348 | . 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒))) |
5 | df-nf 1792 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓)) | |
6 | df-nf 1792 | . 2 ⊢ (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒)) | |
7 | 4, 5, 6 | 3bitr4g 317 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 ∃wex 1787 Ⅎwnf 1791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 |
This theorem depends on definitions: df-bi 210 df-ex 1788 df-nf 1792 |
This theorem is referenced by: nfcjust 2878 nfcr 2882 nfcriOLD 2887 nfcriOLDOLD 2888 bj-drnf2v 34678 |
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