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Theorem cbvsbc 3757
 Description: Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker cbvsbcw 3755 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 2872 . . 3 {𝑥𝜑} = {𝑦𝜓}
54eleq2i 2884 . 2 (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑦𝜓})
6 df-sbc 3724 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
7 df-sbc 3724 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
85, 6, 73bitr4i 306 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1785   ∈ wcel 2112  {cab 2779  [wsbc 3723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2382  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-sbc 3724 This theorem is referenced by:  cbvsbcv  3758  cbvcsb  3842
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