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Theorem cbvrexsv 2517
Description: Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
cbvrexsv (x A φy A [y / x]φ)
Distinct variable groups:   x,A   φ,y   y,A
Allowed substitution hint:   φ(x)

Proof of Theorem cbvrexsv
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1397 . . 3 zφ
2 nfs1v 1791 . . 3 x[z / x]φ
3 sbequ12 1630 . . 3 (x = z → (φ ↔ [z / x]φ))
41, 2, 3cbvrex 2502 . 2 (x A φz A [z / x]φ)
5 nfv 1397 . . . 4 yφ
65nfsb 1798 . . 3 y[z / x]φ
7 nfv 1397 . . 3 z[y / x]φ
8 sbequ 1697 . . 3 (z = y → ([z / x]φ ↔ [y / x]φ))
96, 7, 8cbvrex 2502 . 2 (z A [z / x]φy A [y / x]φ)
104, 9bitri 173 1 (x A φy A [y / x]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  [wsb 1621  wrex 2279
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-io 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1622  df-cleq 2009  df-clel 2012  df-nfc 2143  df-rex 2284
This theorem is referenced by:  rspesbca  2813  rexxpf  4401
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