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Theorem cbvsbc 3812
Description: Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker cbvsbcw 3810 when possible. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.)
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 2801 . . 3 {𝑥𝜑} = {𝑦𝜓}
54eleq2i 2818 . 2 (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑦𝜓})
6 df-sbc 3777 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
7 df-sbc 3777 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
85, 6, 73bitr4i 302 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wnf 1778  wcel 2099  {cab 2703  [wsbc 3776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2366  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sbc 3777
This theorem is referenced by:  cbvsbcv  3813  cbvcsb  3903
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