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Theorem cbvmo 1918
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvmo.1 yφ
cbvmo.2 xψ
cbvmo.3 (x = y → (φψ))
Assertion
Ref Expression
cbvmo (∃*xφ∃*yψ)

Proof of Theorem cbvmo
StepHypRef Expression
1 cbvmo.1 . . . 4 yφ
2 cbvmo.2 . . . 4 xψ
3 cbvmo.3 . . . 4 (x = y → (φψ))
41, 2, 3cbvex 1617 . . 3 (xφyψ)
51, 2, 3cbveu 1902 . . 3 (∃!xφ∃!yψ)
64, 5imbi12i 228 . 2 ((xφ∃!xφ) ↔ (yψ∃!yψ))
7 df-mo 1882 . 2 (∃*xφ ↔ (xφ∃!xφ))
8 df-mo 1882 . 2 (∃*yψ ↔ (yψ∃!yψ))
96, 7, 83bitr4i 201 1 (∃*xφ∃*yψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wnf 1325  wex 1358  ∃!weu 1878  ∃*wmo 1879
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-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882
This theorem is referenced by:  dffun6f  4837
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