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Theorem cbvsbcvw2 36173
Description: Change bound variable of a class substitution using implicit substitution. General version of cbvsbcvw 3826. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
cbvsbcvw2.1 𝐴 = 𝐵
cbvsbcvw2.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvsbcvw2 ([𝐴 / 𝑥]𝜑[𝐵 / 𝑦]𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem cbvsbcvw2
StepHypRef Expression
1 cbvsbcvw2.1 . . 3 𝐴 = 𝐵
2 cbvsbcvw2.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32cbvabv 2808 . . 3 {𝑥𝜑} = {𝑦𝜓}
41, 3eleq12i 2830 . 2 (𝐴 ∈ {𝑥𝜑} ↔ 𝐵 ∈ {𝑦𝜓})
5 df-sbc 3792 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
6 df-sbc 3792 . 2 ([𝐵 / 𝑦]𝜓𝐵 ∈ {𝑦𝜓})
74, 5, 63bitr4i 303 1 ([𝐴 / 𝑥]𝜑[𝐵 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1535  wcel 2104  {cab 2710  [wsbc 3791
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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-sbc 3792
This theorem is referenced by:  cbvcsbvw2  36174
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