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Theorem cbv2h 1631
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
cbv2h.1 (φ → (ψyψ))
cbv2h.2 (φ → (χxχ))
cbv2h.3 (φ → (x = y → (ψχ)))
Assertion
Ref Expression
cbv2h (xyφ → (xψyχ))

Proof of Theorem cbv2h
StepHypRef Expression
1 cbv2h.1 . . 3 (φ → (ψyψ))
2 cbv2h.2 . . 3 (φ → (χxχ))
3 cbv2h.3 . . . 4 (φ → (x = y → (ψχ)))
4 bi1 111 . . . 4 ((ψχ) → (ψχ))
53, 4syl6 29 . . 3 (φ → (x = y → (ψχ)))
61, 2, 5cbv1h 1630 . 2 (xyφ → (xψyχ))
7 equcomi 1589 . . . . 5 (y = xx = y)
8 bi2 121 . . . . 5 ((ψχ) → (χψ))
97, 3, 8syl56 30 . . . 4 (φ → (y = x → (χψ)))
102, 1, 9cbv1h 1630 . . 3 (yxφ → (yχxψ))
1110a7s 1340 . 2 (xyφ → (yχxψ))
126, 11impbid 120 1 (xyφ → (xψyχ))
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  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  cbv2  1632
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