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Theorem dral1 2474
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 2189. (Revised by Wolf Lammen, 6-Sep-2018.)
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 2183 . . 3 𝑥𝑥 𝑥 = 𝑦
2 dral1.1 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albid 2245 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4 axc11 2465 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
5 axc11r 2348 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓))
64, 5impbid 202 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓))
73, 6bitrd 268 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-12 2202  ax-13 2407
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-ex 1852  df-nf 1857
This theorem is referenced by:  drex1  2476  drnf1  2478  axc16gALT  2513  sb9  2572  ralcom2  3251  axpownd  9624  wl-dral1d  33646  wl-ax11-lem5  33693  wl-ax11-lem8  33696  wl-ax11-lem9  33697
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