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Theorem cbvrexsv 2539
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 1418 . . 3 zφ
2 nfs1v 1812 . . 3 x[z / x]φ
3 sbequ12 1651 . . 3 (x = z → (φ ↔ [z / x]φ))
41, 2, 3cbvrex 2524 . 2 (x A φz A [z / x]φ)
5 nfv 1418 . . . 4 yφ
65nfsb 1819 . . 3 y[z / x]φ
7 nfv 1418 . . 3 z[y / x]φ
8 sbequ 1718 . . 3 (z = y → ([z / x]φ ↔ [y / x]φ))
96, 7, 8cbvrex 2524 . 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 1642  wrex 2301
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306
This theorem is referenced by:  rspesbca  2836  rexxpf  4426
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