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Theorem cbv3 1603
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3.1 yφ
cbv3.2 xψ
cbv3.3 (x = y → (φψ))
Assertion
Ref Expression
cbv3 (xφyψ)

Proof of Theorem cbv3
StepHypRef Expression
1 cbv3.1 . . 3 yφ
21nfal 1441 . 2 yxφ
3 cbv3.2 . . 3 xψ
4 cbv3.3 . . 3 (x = y → (φψ))
53, 4spim 1599 . 2 (xφψ)
62, 5alrimi 1388 1 (xφyψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1221  wnf 1322
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 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-4 1373  ax-i9 1396  ax-ial 1400
This theorem depends on definitions:  df-bi 110  df-nf 1323
This theorem is referenced by:  cbv3h  1604  cbv1  1605  mo2n  1901  mo23  1914  setindis  8347  bdsetindis  8349
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