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Theorem cbvab 2176
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 1838 . . . 4 𝑥[𝑧 / 𝑦]𝜓
3 cbvab.1 . . . . . 6 𝑦𝜑
4 cbvab.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
54equcoms 1610 . . . . . . 7 (𝑦 = 𝑥 → (𝜑𝜓))
65bicomd 133 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
73, 6sbie 1690 . . . . 5 ([𝑥 / 𝑦]𝜓𝜑)
8 sbequ 1737 . . . . 5 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜓 ↔ [𝑧 / 𝑦]𝜓))
97, 8syl5bbr 187 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜓))
102, 9sbie 1690 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
11 df-clab 2043 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
12 df-clab 2043 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
1310, 11, 123bitr4i 205 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1413eqriv 2053 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wnf 1365  wcel 1409  [wsb 1661  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049
This theorem is referenced by:  cbvabv  2177  cbvrab  2572  cbvsbc  2813  cbvrabcsf  2938  dfdmf  4555  dfrnf  4602  funfvdm2f  5265  abrexex2g  5774  abrexex2  5778
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