MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  aev2 Structured version   Visualization version   GIF version

Theorem aev2 2061
Description: A version of aev 2060 with two universal quantifiers in the consequent. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 2059, aev 2060, aev2 2061).

Using aev 2060 and alrimiv 1930, one can actually prove (with no more axioms) any scheme of the form (∀𝑥𝑥 = 𝑦 PHI) , DV (𝑥, 𝑦) where PHI involves only setvar variables and the connectors , , , , , =, , , ∃*, ∃!, . An example is given by aevdemo 28824. This list cannot be extended to ¬ or since the scheme 𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1798, ax-1 6-- ax-13 2372 (as the one-element universe shows), so for instance (∀𝑥𝑥 = 𝑦 → ⊥), DV (𝑥, 𝑦) is not provable from these axioms alone (indeed, dtru 5359 uses non-logical axioms as well). (Contributed by BJ, 23-Mar-2021.)

Assertion
Ref Expression
aev2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)
Distinct variable group:   𝑥,𝑦

Proof of Theorem aev2
Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aev 2060 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑤 𝑤 = 𝑠)
2 aev 2060 . . 3 (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣)
32alrimiv 1930 . 2 (∀𝑤 𝑤 = 𝑠 → ∀𝑧𝑡 𝑢 = 𝑣)
41, 3syl 17 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783
This theorem is referenced by:  hbaev  2062
  Copyright terms: Public domain W3C validator