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Theorem ax11v 1705
 Description: This is a version of ax-11o 1701 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)
Assertion
Ref Expression
ax11v (x = y → (φx(x = yφ)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11v
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 a9e 1583 . 2 z z = y
2 ax-17 1416 . . . . 5 (φzφ)
3 ax-11 1394 . . . . 5 (x = z → (zφx(x = zφ)))
42, 3syl5 28 . . . 4 (x = z → (φx(x = zφ)))
5 equequ2 1596 . . . . 5 (z = y → (x = zx = y))
65imbi1d 220 . . . . . . 7 (z = y → ((x = zφ) ↔ (x = yφ)))
76albidv 1702 . . . . . 6 (z = y → (x(x = zφ) ↔ x(x = yφ)))
87imbi2d 219 . . . . 5 (z = y → ((φx(x = zφ)) ↔ (φx(x = yφ))))
95, 8imbi12d 223 . . . 4 (z = y → ((x = z → (φx(x = zφ))) ↔ (x = y → (φx(x = yφ)))))
104, 9mpbii 136 . . 3 (z = y → (x = y → (φx(x = yφ))))
1110exlimiv 1486 . 2 (z z = y → (x = y → (φx(x = yφ))))
121, 11ax-mp 7 1 (x = y → (φx(x = yφ)))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   = wceq 1242  ∃wex 1378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-17 1416  ax-i9 1420 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  equs5or  1708  sb56  1762
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