MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax12wdemo Structured version   Visualization version   GIF version

Theorem ax12wdemo 2052
Description: Example of an application of ax12w 2050 that results in an instance of ax-12 2087 for a contrived formula with mixed free and bound variables, (𝑥𝑦 ∧ ∀𝑥𝑧𝑥 ∧ ∀𝑦𝑧𝑦𝑥), in place of 𝜑. The proof illustrates bound variable renaming with cbvalvw 2011 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)
Assertion
Ref Expression
ax12wdemo (𝑥 = 𝑦 → (∀𝑦(𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥))))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem ax12wdemo
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elequ1 2037 . . 3 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
2 elequ2 2044 . . . . 5 (𝑥 = 𝑤 → (𝑧𝑥𝑧𝑤))
32cbvalvw 2011 . . . 4 (∀𝑥 𝑧𝑥 ↔ ∀𝑤 𝑧𝑤)
43a1i 11 . . 3 (𝑥 = 𝑦 → (∀𝑥 𝑧𝑥 ↔ ∀𝑤 𝑧𝑤))
5 elequ1 2037 . . . . . 6 (𝑦 = 𝑣 → (𝑦𝑥𝑣𝑥))
65albidv 1889 . . . . 5 (𝑦 = 𝑣 → (∀𝑧 𝑦𝑥 ↔ ∀𝑧 𝑣𝑥))
76cbvalvw 2011 . . . 4 (∀𝑦𝑧 𝑦𝑥 ↔ ∀𝑣𝑧 𝑣𝑥)
8 elequ2 2044 . . . . . 6 (𝑥 = 𝑦 → (𝑣𝑥𝑣𝑦))
98albidv 1889 . . . . 5 (𝑥 = 𝑦 → (∀𝑧 𝑣𝑥 ↔ ∀𝑧 𝑣𝑦))
109albidv 1889 . . . 4 (𝑥 = 𝑦 → (∀𝑣𝑧 𝑣𝑥 ↔ ∀𝑣𝑧 𝑣𝑦))
117, 10syl5bb 272 . . 3 (𝑥 = 𝑦 → (∀𝑦𝑧 𝑦𝑥 ↔ ∀𝑣𝑧 𝑣𝑦))
121, 4, 113anbi123d 1439 . 2 (𝑥 = 𝑦 → ((𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥) ↔ (𝑦𝑦 ∧ ∀𝑤 𝑧𝑤 ∧ ∀𝑣𝑧 𝑣𝑦)))
13 elequ2 2044 . . 3 (𝑦 = 𝑣 → (𝑥𝑦𝑥𝑣))
147a1i 11 . . 3 (𝑦 = 𝑣 → (∀𝑦𝑧 𝑦𝑥 ↔ ∀𝑣𝑧 𝑣𝑥))
1513, 143anbi13d 1441 . 2 (𝑦 = 𝑣 → ((𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥) ↔ (𝑥𝑣 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑣𝑧 𝑣𝑥)))
1612, 15ax12w 2050 1 (𝑥 = 𝑦 → (∀𝑦(𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1056  df-ex 1745
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator