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Theorem cbvsbcw 2964
 Description: Version of cbvsbc 2965 with a disjoint variable condition. (Contributed 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 2280 . . 3 {𝑥𝜑} = {𝑦𝜓}
54eleq2i 2224 . 2 (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑦𝜓})
6 df-sbc 2938 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
7 df-sbc 2938 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
85, 6, 73bitr4i 211 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1440   ∈ wcel 2128  {cab 2143  [wsbc 2937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-sbc 2938 This theorem is referenced by:  cbvcsbw  3035
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