Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dral1v Structured version   Visualization version   GIF version

Theorem dral1v 2376
 Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2450 with a disjoint variable condition, which does not require ax-13 2379. Remark: the corresponding versions for dral2 2449 and drex2 2453 are instances of albidv 1921 and exbidv 1922 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2176. (Revised by Wolf Lammen, 30-Mar-2024.)
Hypothesis
Ref Expression
dral1v.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dral1v (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem dral1v
StepHypRef Expression
1 nfa1 2152 . . 3 𝑥𝑥 𝑥 = 𝑦
2 dral1v.1 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albid 2222 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4 axc11v 2262 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
5 axc11rv 2263 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓))
64, 5impbid 215 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓))
73, 6bitrd 282 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536 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-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  drex1v  2377  drnf1v  2378
 Copyright terms: Public domain W3C validator