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Theorem cbvab 2237
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 1897 . . . 4 𝑥[𝑧 / 𝑦]𝜓
3 cbvab.1 . . . . . 6 𝑦𝜑
4 cbvab.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
54equcoms 1667 . . . . . . 7 (𝑦 = 𝑥 → (𝜑𝜓))
65bicomd 140 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
73, 6sbie 1747 . . . . 5 ([𝑥 / 𝑦]𝜓𝜑)
8 sbequ 1794 . . . . 5 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜓 ↔ [𝑧 / 𝑦]𝜓))
97, 8syl5bbr 193 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜓))
102, 9sbie 1747 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
11 df-clab 2102 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
12 df-clab 2102 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
1310, 11, 123bitr4i 211 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1413eqriv 2112 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wnf 1419  wcel 1463  [wsb 1718  {cab 2101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108
This theorem is referenced by:  cbvabv  2238  cbvrab  2655  cbvsbc  2905  cbvrabcsf  3031  dfdmf  4692  dfrnf  4740  funfvdm2f  5440  abrexex2g  5972  abrexex2  5976
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