Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dral2 Structured version   Visualization version   GIF version

Theorem dral2 2462
 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 2392. Usage of albidv 1922 is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005.) Allow a shortening of dral1 2463. (Revised by Wolf Lammen, 4-Mar-2018.) (New usage is discouraged.)
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 2457 . 2 𝑧𝑥 𝑥 = 𝑦
2 dral1.1 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albid 2226 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
 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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by:  dral1ALT  2464  sbal1  2574  sbal2  2575  sbal2OLD  2576  axpownd  10010  wl-sbalnae  34868
 Copyright terms: Public domain W3C validator