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Theorem cbvab 2142
 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 {xφ} = {yψ}

Proof of Theorem cbvab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 cbvab.2 . . . . 5 xψ
21nfsb 1804 . . . 4 x[z / y]ψ
3 cbvab.1 . . . . . 6 yφ
4 cbvab.3 . . . . . . . 8 (x = y → (φψ))
54equcoms 1576 . . . . . . 7 (y = x → (φψ))
65bicomd 129 . . . . . 6 (y = x → (ψφ))
73, 6sbie 1656 . . . . 5 ([x / y]ψφ)
8 sbequ 1703 . . . . 5 (x = z → ([x / y]ψ ↔ [z / y]ψ))
97, 8syl5bbr 183 . . . 4 (x = z → (φ ↔ [z / y]ψ))
102, 9sbie 1656 . . 3 ([z / x]φ ↔ [z / y]ψ)
11 df-clab 2009 . . 3 (z {xφ} ↔ [z / x]φ)
12 df-clab 2009 . . 3 (z {yψ} ↔ [z / y]ψ)
1310, 11, 123bitr4i 201 . 2 (z {xφ} ↔ z {yψ})
1413eqriv 2019 1 {xφ} = {yψ}
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228  Ⅎwnf 1329   ∈ wcel 1374  [wsb 1627  {cab 2008 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015 This theorem is referenced by:  cbvabv  2143  cbvrab  2533  cbvsbc  2768  cbvrabcsf  2888  dfdmf  4455  dfrnf  4502  funfvdm2f  5163  abrexex2g  5670  abrexex2  5674
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