 Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax12fromc15 Structured version   Visualization version   GIF version

Theorem ax12fromc15 35059
 Description: Rederivation of axiom ax-12 2163 from ax-c15 35043, ax-c11 35041 (used through dral1-o 35058), and other older axioms. See theorem axc15 2387 for the derivation of ax-c15 35043 from ax-12 2163. An open problem is whether we can prove this using ax-c11n 35042 instead of ax-c11 35041. This proof uses newer axioms ax-4 1853 and ax-6 2021, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 35038 and ax-c10 35040. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax12fromc15 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem ax12fromc15
StepHypRef Expression
1 biidd 254 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜑))
21dral1-o 35058 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
3 ax-1 6 . . . . 5 (𝜑 → (𝑥 = 𝑦𝜑))
43alimi 1855 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
52, 4syl6bir 246 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
65a1d 25 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
7 ax-c5 35037 . . 3 (∀𝑦𝜑𝜑)
8 ax-c15 35043 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
97, 8syl7 74 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
106, 9pm2.61i 177 1 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-11 2150  ax-c5 35037  ax-c4 35038  ax-c7 35039  ax-c10 35040  ax-c11 35041  ax-c15 35043  ax-c9 35044 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator