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Mirrors > Home > MPE Home > Th. List > dral1 | Structured version Visualization version GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.) Remove dependency on ax-11 2126. (Revised by Wolf Lammen, 6-Sep-2018.) |
Ref | Expression |
---|---|
dral1.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dral1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2121 | . . 3 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦 | |
2 | dral1.1 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | albid 2189 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
4 | axc11 2409 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓)) | |
5 | axc11r 2343 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓)) | |
6 | 4, 5 | impbid 213 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓)) |
7 | 3, 6 | bitrd 280 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-10 2112 ax-12 2141 ax-13 2344 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ex 1762 df-nf 1766 |
This theorem is referenced by: drex1 2420 drnf1 2422 axc16gALT 2483 sb9 2515 ralcom2 3324 axpownd 9869 wl-dral1d 34303 wl-ax11-lem5 34352 wl-ax11-lem8 34355 wl-ax11-lem9 34356 |
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