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

Theorem aevdemo 28242
Description: Proof illustrating the comment of aev2 2062. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
aevdemo (∀𝑥 𝑥 = 𝑦 → ((∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀(𝑖 = 𝑗𝑘 = 𝑙)))
Distinct variable group:   𝑥,𝑦

Proof of Theorem aevdemo
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aev 2061 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑒 𝑓 = 𝑔)
2119.2d 1981 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑒 𝑓 = 𝑔)
32olcd 870 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔))
4 aev 2061 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑚 𝑚 = 𝑛)
5 aeveq 2060 . . . . 5 (∀𝑚 𝑚 = 𝑛𝑘 = 𝑙)
65a1d 25 . . . 4 (∀𝑚 𝑚 = 𝑛 → (𝑖 = 𝑗𝑘 = 𝑙))
76alrimiv 1927 . . 3 (∀𝑚 𝑚 = 𝑛 → ∀(𝑖 = 𝑗𝑘 = 𝑙))
84, 7syl 17 . 2 (∀𝑥 𝑥 = 𝑦 → ∀(𝑖 = 𝑗𝑘 = 𝑙))
93, 8jca 514 1 (∀𝑥 𝑥 = 𝑦 → ((∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀(𝑖 = 𝑗𝑘 = 𝑙)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843  wal 1534  wex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator