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Theorem cbvsbcw 3755
Description: Change bound variables in a wff substitution. Version of cbvsbc 3757 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by Jeff Hankins, 19-Sep-2009.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvsbcw.1 𝑦𝜑
cbvsbcw.2 𝑥𝜓
cbvsbcw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvsbcw ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem cbvsbcw
StepHypRef Expression
1 cbvsbcw.1 . . . 4 𝑦𝜑
2 cbvsbcw.2 . . . 4 𝑥𝜓
3 cbvsbcw.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvabw 2870 . . 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  wnf 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-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  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:  cbvsbcvw  3756  cbvcsbw  3841
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