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Theorem cbvsbc 3451
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 2749 . . 3 {𝑥𝜑} = {𝑦𝜓}
54eleq2i 2696 . 2 (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑦𝜓})
6 df-sbc 3423 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
7 df-sbc 3423 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
85, 6, 73bitr4i 292 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1705  wcel 1992  {cab 2612  [wsbc 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-sbc 3423
This theorem is referenced by:  cbvsbcv  3452  cbvcsb  3524
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