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Theorem ax12wdemo 1831
 Description: Example of an application of ax12w 1829 that results in an instance of ax-12 1854 for a contrived formula with mixed free and bound variables, , in place of . The proof illustrates bound variable renaming with cbvalvw 1809 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
Distinct variable group:   ,,

Proof of Theorem ax12wdemo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elequ1 1821 . . 3
2 elequ2 1823 . . . . 5
32cbvalvw 1809 . . . 4
43a1i 11 . . 3
5 elequ1 1821 . . . . . 6
65albidv 1713 . . . . 5
76cbvalvw 1809 . . . 4
8 elequ2 1823 . . . . . 6
98albidv 1713 . . . . 5
109albidv 1713 . . . 4
117, 10syl5bb 257 . . 3
121, 4, 113anbi123d 1299 . 2
13 elequ2 1823 . . 3
147a1i 11 . . 3
1513, 143anbi13d 1301 . 2
1612, 15ax12w 1829 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\w3a 973  A.wal 1393 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822 This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1613
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