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Theorem cbvab 2471
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
cbvab.1 yφ
cbvab.2 xψ
cbvab.3 (x = y → (φψ))
Assertion
Ref Expression
cbvab {x φ} = {y ψ}

Proof of Theorem cbvab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 cbvab.2 . . . . 5 xψ
21nfsb 2109 . . . 4 x[z / y]ψ
3 cbvab.1 . . . . . 6 yφ
4 cbvab.3 . . . . . . . 8 (x = y → (φψ))
54equcoms 1681 . . . . . . 7 (y = x → (φψ))
65bicomd 192 . . . . . 6 (y = x → (ψφ))
73, 6sbie 2038 . . . . 5 ([x / y]ψφ)
8 sbequ 2060 . . . . 5 (x = z → ([x / y]ψ ↔ [z / y]ψ))
97, 8syl5bbr 250 . . . 4 (x = z → (φ ↔ [z / y]ψ))
102, 9sbie 2038 . . 3 ([z / x]φ ↔ [z / y]ψ)
11 df-clab 2340 . . 3 (z {x φ} ↔ [z / x]φ)
12 df-clab 2340 . . 3 (z {y ψ} ↔ [z / y]ψ)
1310, 11, 123bitr4i 268 . 2 (z {x φ} ↔ z {y ψ})
1413eqriv 2350 1 {x φ} = {y ψ}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 This theorem is referenced by:  cbvabv  2472  cbvrab  2857  cbvsbc  3074  cbvrabcsf  3201  dfdmf  4905  dfrnf  4962  funfv2f  5377
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