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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 2021. (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 2015 | . . 3 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦 | |
2 | dral1.1 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | albid 2077 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
4 | axc11 2302 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓)) | |
5 | axc11r 2175 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓)) | |
6 | 4, 5 | impbid 201 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓)) |
7 | 3, 6 | bitrd 267 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: drex1 2315 drnf1 2317 ax12OLD 2329 axc16gALT 2355 sb9 2414 ralcom2 3083 axpownd 9302 wl-dral1d 32497 wl-ax11-lem5 32545 wl-ax11-lem8 32548 wl-ax11-lem9 32549 |
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