 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc15 Structured version   Visualization version   GIF version

Theorem axc15 2387
 Description: Derivation of set.mm's original ax-c15 35052 from ax-c11n 35051 and the shorter ax-12 2163 that has replaced it. Theorem ax12 2389 shows the reverse derivation of ax-12 2163 from ax-c15 35052. Normally, axc15 2387 should be used rather than ax-c15 35052, except by theorems specifically studying the latter's properties. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 26-Mar-2023.)
Assertion
Ref Expression
axc15 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Proof of Theorem axc15
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 2023 . 2 𝑧 𝑧 = 𝑦
2 dveeq2 2342 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3 ax12v 2164 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
4 equeuclr 2070 . . . . . 6 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
54sps 2169 . . . . 5 (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
64imim1d 82 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑦𝜑)))
76al2imi 1859 . . . . . 6 (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
87imim2d 57 . . . . 5 (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
95, 8imim12d 81 . . . 4 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
102, 3, 9syl6mpi 67 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
1110exlimdv 1976 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
121, 11mpi 20 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599  ∃wex 1823 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-10 2135  ax-12 2163  ax-13 2334 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828 This theorem is referenced by:  ax12  2389  ax12b  2390  equs5  2426  ax12vALT  2505  bj-ax12v3ALT  33273
 Copyright terms: Public domain W3C validator