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Mirrors > Home > MPE Home > Th. List > drnf1v | Structured version Visualization version GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2438 with a disjoint variable condition, which does not require ax-13 2367. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2130. (Revised by Gino Giotto, 18-Nov-2024.) |
Ref | Expression |
---|---|
dral1v.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
drnf1v | ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dral1v.1 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | drex1v 2364 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓)) |
3 | 1 | dral1v 2362 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
4 | 2, 3 | imbi12d 344 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑦𝜓 → ∀𝑦𝜓))) |
5 | df-nf 1779 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
6 | df-nf 1779 | . 2 ⊢ (Ⅎ𝑦𝜓 ↔ (∃𝑦𝜓 → ∀𝑦𝜓)) | |
7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 ∃wex 1774 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 |
This theorem is referenced by: nfriotadw 7384 |
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