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Theorem dral2 2323
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, 27-Feb-2005.) Allow a shortening of dral1 2324. (Revised by Wolf Lammen, 4-Mar-2018.)
Hypothesis
Ref Expression
dral1.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dral2 (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Proof of Theorem dral2
StepHypRef Expression
1 nfae 2315 . 2 𝑧𝑥 𝑥 = 𝑦
2 dral1.1 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albid 2088 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by:  dral1ALT  2325  sbal1  2459  sbal2  2460  drnfc1  2778  drnfc2  2779  axpownd  9383  wl-sbalnae  33016
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