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Theorem cbvrexsv 2519
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 1398 . . 3 zφ
2 nfs1v 1793 . . 3 x[z / x]φ
3 sbequ12 1632 . . 3 (x = z → (φ ↔ [z / x]φ))
41, 2, 3cbvrex 2504 . 2 (x A φz A [z / x]φ)
5 nfv 1398 . . . 4 yφ
65nfsb 1800 . . 3 y[z / x]φ
7 nfv 1398 . . 3 z[y / x]φ
8 sbequ 1699 . . 3 (z = y → ([z / x]φ ↔ [y / x]φ))
96, 7, 8cbvrex 2504 . 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 1623  wrex 2281
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286
This theorem is referenced by:  rspesbca  2815  rexxpf  4406
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