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Theorem cbvab 2301
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 1946 . . . 4 𝑥[𝑧 / 𝑦]𝜓
3 cbvab.1 . . . . . 6 𝑦𝜑
4 cbvab.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
54equcoms 1708 . . . . . . 7 (𝑦 = 𝑥 → (𝜑𝜓))
65bicomd 141 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
73, 6sbie 1791 . . . . 5 ([𝑥 / 𝑦]𝜓𝜑)
8 sbequ 1840 . . . . 5 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜓 ↔ [𝑧 / 𝑦]𝜓))
97, 8bitr3id 194 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜓))
102, 9sbie 1791 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
11 df-clab 2164 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
12 df-clab 2164 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
1310, 11, 123bitr4i 212 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1413eqriv 2174 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wnf 1460  [wsb 1762  wcel 2148  {cab 2163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170
This theorem is referenced by:  cbvabv  2302  cbvrab  2735  cbvsbc  2991  cbvrabcsf  3122  dfdmf  4820  dfrnf  4868  funfvdm2f  5581  abrexex2g  6120  abrexex2  6124
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