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Theorem cbvsbcdavw2 36201
Description: Change bound variable of a class substitution. General version of cbvsbcdavw 36200. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvsbcdavw2.1 (𝜑𝐴 = 𝐵)
cbvsbcdavw2.2 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvsbcdavw2 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑦]𝜒))
Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem cbvsbcdavw2
StepHypRef Expression
1 cbvsbcdavw2.1 . . 3 (𝜑𝐴 = 𝐵)
2 cbvsbcdavw2.2 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
32cbvabdavw 36199 . . 3 (𝜑 → {𝑥𝜓} = {𝑦𝜒})
41, 3eleq12d 2831 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐵 ∈ {𝑦𝜒}))
5 df-sbc 3792 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 3792 . 2 ([𝐵 / 𝑦]𝜒𝐵 ∈ {𝑦𝜒})
74, 5, 63bitr4g 314 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑦]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = 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:  cbvcsbdavw2  36203
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