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Theorem ax11v 1686
Description: This is a version of ax-11o 1682 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 1564 . 2 z z = y
2 ax-17 1396 . . . . 5 (φzφ)
3 ax-11 1374 . . . . 5 (x = z → (zφx(x = zφ)))
42, 3syl5 28 . . . 4 (x = z → (φx(x = zφ)))
5 equequ2 1577 . . . . 5 (z = y → (x = zx = y))
65imbi1d 220 . . . . . . 7 (z = y → ((x = zφ) ↔ (x = yφ)))
76albidv 1683 . . . . . 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 1467 . 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 1224   = wceq 1226  wex 1358
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 1312  ax-gen 1314  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-17 1396  ax-i9 1400
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  equs5or  1689  sb56  1743
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