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Theorem cbvexh 1635
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
cbvexh.1 (φyφ)
cbvexh.2 (ψxψ)
cbvexh.3 (x = y → (φψ))
Assertion
Ref Expression
cbvexh (xφyψ)

Proof of Theorem cbvexh
StepHypRef Expression
1 cbvexh.2 . . . 4 (ψxψ)
21hbex 1524 . . 3 (yψxyψ)
3 cbvexh.1 . . . . 5 (φyφ)
4 cbvexh.3 . . . . . . 7 (x = y → (φψ))
54bicomd 129 . . . . . 6 (x = y → (ψφ))
65equcoms 1591 . . . . 5 (y = x → (ψφ))
73, 6equsex 1613 . . . 4 (y(y = x ψ) ↔ φ)
8 simpr 103 . . . . 5 ((y = x ψ) → ψ)
98eximi 1488 . . . 4 (y(y = x ψ) → yψ)
107, 9sylbir 125 . . 3 (φyψ)
112, 10exlimih 1481 . 2 (xφyψ)
123hbex 1524 . . 3 (xφyxφ)
131, 4equsex 1613 . . . 4 (x(x = y φ) ↔ ψ)
14 simpr 103 . . . . 5 ((x = y φ) → φ)
1514eximi 1488 . . . 4 (x(x = y φ) → xφ)
1613, 15sylbir 125 . . 3 (ψxφ)
1712, 16exlimih 1481 . 2 (yψxφ)
1811, 17impbii 117 1 (xφyψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378
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
This theorem is referenced by:  cbvex  1636  sb8eh  1732  cbvexv  1792  euf  1902  mopick  1975
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