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Theorem cbvsbc 2791
Description: Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvsbc.1 𝑦𝜑
cbvsbc.2 𝑥𝜓
cbvsbc.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvsbc ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)

Proof of Theorem cbvsbc
StepHypRef Expression
1 cbvsbc.1 . . . 4 𝑦𝜑
2 cbvsbc.2 . . . 4 𝑥𝜓
3 cbvsbc.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvab 2160 . . 3 {𝑥𝜑} = {𝑦𝜓}
54eleq2i 2104 . 2 (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑦𝜓})
6 df-sbc 2765 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
7 df-sbc 2765 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
85, 6, 73bitr4i 201 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wnf 1349  wcel 1393  {cab 2026  [wsbc 2764
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765
This theorem is referenced by:  cbvsbcv  2792  cbvcsb  2856
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