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Theorem cbvsbc 3074
 Description: Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvsbc.1 yφ
cbvsbc.2 xψ
cbvsbc.3 (x = y → (φψ))
Assertion
Ref Expression
cbvsbc ([̣A / xφ ↔ [̣A / yψ)

Proof of Theorem cbvsbc
StepHypRef Expression
1 cbvsbc.1 . . . 4 yφ
2 cbvsbc.2 . . . 4 xψ
3 cbvsbc.3 . . . 4 (x = y → (φψ))
41, 2, 3cbvab 2471 . . 3 {x φ} = {y ψ}
54eleq2i 2417 . 2 (A {x φ} ↔ A {y ψ})
6 df-sbc 3047 . 2 ([̣A / xφA {x φ})
7 df-sbc 3047 . 2 ([̣A / yψA {y ψ})
85, 6, 73bitr4i 268 1 ([̣A / xφ ↔ [̣A / yψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339  [̣wsbc 3046 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-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047 This theorem is referenced by:  cbvsbcv  3075  cbvcsb  3140
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