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Theorem cbvralsv 2846
 Description: Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
cbvralsv (x A φy A [y / x]φ)
Distinct variable groups:   x,A   y,A   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem cbvralsv
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . 3 zφ
2 nfs1v 2106 . . 3 x[z / x]φ
3 sbequ12 1919 . . 3 (x = z → (φ ↔ [z / x]φ))
41, 2, 3cbvral 2831 . 2 (x A φz A [z / x]φ)
5 nfv 1619 . . . 4 yφ
65nfsb 2109 . . 3 y[z / x]φ
7 nfv 1619 . . 3 z[y / x]φ
8 sbequ 2060 . . 3 (z = y → ([z / x]φ ↔ [y / x]φ))
96, 7, 8cbvral 2831 . 2 (z A [z / x]φy A [y / x]φ)
104, 9bitri 240 1 (x A φy A [y / x]φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  [wsb 1648  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619 This theorem is referenced by:  sbralie  2848  rspsbc  3124  ralxpf  4827
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