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Theorem dral1v 2358
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2430 with a disjoint variable condition, which does not require ax-13 2363. Remark: the corresponding versions for dral2 2429 and drex2 2433 are instances of albidv 1915 and exbidv 1916 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2164. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2129. (Revised by Gino Giotto, 18-Nov-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 hbaev 2054 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
2 dral1v.1 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albidh 1861 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4 axc11v 2247 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
5 axc11rv 2248 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓))
64, 5impbid 211 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓))
73, 6bitrd 279 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774
This theorem is referenced by:  drex1v  2360  drnf1v  2361  drnf1vOLD  2362
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