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Theorem dral1 2450
 Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker dral1v 2376 if possible. (Contributed by NM, 24-Nov-1994.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)
Hypothesis
Ref Expression
dral1.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dral1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Proof of Theorem dral1
StepHypRef Expression
1 nfa1 2152 . . 3 𝑥𝑥 𝑥 = 𝑦
2 dral1.1 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albid 2222 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4 axc11 2441 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
5 axc11r 2375 . . 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  ax-13 2379 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:  drex1  2452  drnf1  2454  axc16gALT  2508  sb9  2538  ralcom2  3281  axpownd  10061  wl-dral1d  35216  wl-ax11-lem5  35266  wl-ax11-lem8  35269  wl-ax11-lem9  35270
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