 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax12vOLDOLD Structured version   Visualization version   GIF version

Theorem ax12vOLDOLD 2038
 Description: Obsolete proof of ax12v 2035 as of 7-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2006 and ax-13 2234. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax12vOLDOLD (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12vOLDOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1940 . . . 4 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
21biimprd 237 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
3 ax-5 1827 . . . . 5 (𝜑 → ∀𝑧𝜑)
4 ax-12 2034 . . . . 5 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
53, 4syl5 33 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
62imim1d 80 . . . . 5 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑦𝜑)))
76alimdv 1832 . . . 4 (𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
85, 7syl9r 76 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
92, 8syld 46 . 2 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
10 ax6ev 1877 . 2 𝑧 𝑧 = 𝑦
119, 10exlimiiv 1846 1 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator