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Theorem cbv2 1613
Description: Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv2.1 xφ
cbv2.2 yφ
cbv2.3 (φ → Ⅎyψ)
cbv2.4 (φ → Ⅎxχ)
cbv2.5 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
cbv2 (φ → (xψyχ))

Proof of Theorem cbv2
StepHypRef Expression
1 cbv2.2 . . . 4 yφ
21nfri 1389 . . 3 (φyφ)
3 cbv2.1 . . . . 5 xφ
43nfal 1446 . . . 4 xyφ
54nfri 1389 . . 3 (yφxyφ)
62, 5syl 14 . 2 (φxyφ)
7 cbv2.3 . . . 4 (φ → Ⅎyψ)
87nfrd 1390 . . 3 (φ → (ψyψ))
9 cbv2.4 . . . 4 (φ → Ⅎxχ)
109nfrd 1390 . . 3 (φ → (χxχ))
11 cbv2.5 . . 3 (φ → (x = y → (ψχ)))
128, 10, 11cbv2h 1612 . 2 (xyφ → (xψyχ))
136, 12syl 14 1 (φ → (xψyχ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1224  wnf 1325
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-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-nf 1326
This theorem is referenced by:  cbvald  1778  cbvrald  7253
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