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Theorem ax12wdemo 1960
Description: Example of an application of ax12w 1958 that results in an instance of ax-12 1983 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 1918 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 1945 . . 3 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
2 elequ2 1952 . . . . 5 (𝑥 = 𝑤 → (𝑧𝑥𝑧𝑤))
32cbvalvw 1918 . . . 4 (∀𝑥 𝑧𝑥 ↔ ∀𝑤 𝑧𝑤)
43a1i 11 . . 3 (𝑥 = 𝑦 → (∀𝑥 𝑧𝑥 ↔ ∀𝑤 𝑧𝑤))
5 elequ1 1945 . . . . . 6 (𝑦 = 𝑣 → (𝑦𝑥𝑣𝑥))
65albidv 1802 . . . . 5 (𝑦 = 𝑣 → (∀𝑧 𝑦𝑥 ↔ ∀𝑧 𝑣𝑥))
76cbvalvw 1918 . . . 4 (∀𝑦𝑧 𝑦𝑥 ↔ ∀𝑣𝑧 𝑣𝑥)
8 elequ2 1952 . . . . . 6 (𝑥 = 𝑦 → (𝑣𝑥𝑣𝑦))
98albidv 1802 . . . . 5 (𝑥 = 𝑦 → (∀𝑧 𝑣𝑥 ↔ ∀𝑧 𝑣𝑦))
109albidv 1802 . . . 4 (𝑥 = 𝑦 → (∀𝑣𝑧 𝑣𝑥 ↔ ∀𝑣𝑧 𝑣𝑦))
117, 10syl5bb 270 . . 3 (𝑥 = 𝑦 → (∀𝑦𝑧 𝑦𝑥 ↔ ∀𝑣𝑧 𝑣𝑦))
121, 4, 113anbi123d 1390 . 2 (𝑥 = 𝑦 → ((𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥) ↔ (𝑦𝑦 ∧ ∀𝑤 𝑧𝑤 ∧ ∀𝑣𝑧 𝑣𝑦)))
13 elequ2 1952 . . 3 (𝑦 = 𝑣 → (𝑥𝑦𝑥𝑣))
147a1i 11 . . 3 (𝑦 = 𝑣 → (∀𝑦𝑧 𝑦𝑥 ↔ ∀𝑣𝑧 𝑣𝑥))
1513, 143anbi13d 1392 . 2 (𝑦 = 𝑣 → ((𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥) ↔ (𝑥𝑣 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑣𝑧 𝑣𝑥)))
1612, 15ax12w 1958 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 194  w3a 1030  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947
This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-ex 1695
This theorem is referenced by: (None)
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