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Theorem cbvalh 1633
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
cbvalh.1 (φyφ)
cbvalh.2 (ψxψ)
cbvalh.3 (x = y → (φψ))
Assertion
Ref Expression
cbvalh (xφyψ)

Proof of Theorem cbvalh
StepHypRef Expression
1 cbvalh.1 . . 3 (φyφ)
2 cbvalh.2 . . 3 (ψxψ)
3 cbvalh.3 . . . 4 (x = y → (φψ))
43biimpd 132 . . 3 (x = y → (φψ))
51, 2, 4cbv3h 1628 . 2 (xφyψ)
63equcoms 1591 . . . 4 (y = x → (φψ))
76biimprd 147 . . 3 (y = x → (ψφ))
82, 1, 7cbv3h 1628 . 2 (yψxφ)
95, 8impbii 117 1 (xφyψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240
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-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  cbval  1634  sb8h  1731  cbvalv  1791  sb9v  1851  sb8euh  1920
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