Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  equidq Structured version   Visualization version   GIF version

Theorem equidq 33728
Description: equid 1936 with universal quantifier without using ax-c5 33687 or ax-5 1836. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equidq 𝑦 𝑥 = 𝑥

Proof of Theorem equidq
StepHypRef Expression
1 equidqe 33726 . 2 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2 ax10fromc7 33699 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3 hbequid 33713 . . . 4 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
43con3i 150 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
52, 4alrimih 1748 . 2 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
61, 5mt3 192 1 𝑦 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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-c5 33687  ax-c4 33688  ax-c7 33689  ax-c10 33690  ax-c9 33694
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator