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Theorem sbceqbidf 30261
 Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
Hypotheses
Ref Expression
sbceqbidf.1 𝑥𝜑
sbceqbidf.2 (𝜑𝐴 = 𝐵)
sbceqbidf.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbceqbidf (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))

Proof of Theorem sbceqbidf
StepHypRef Expression
1 sbceqbidf.2 . . 3 (𝜑𝐴 = 𝐵)
2 sbceqbidf.1 . . . 4 𝑥𝜑
3 sbceqbidf.3 . . . 4 (𝜑 → (𝜓𝜒))
42, 3abbid 2867 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
51, 4eleq12d 2887 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐵 ∈ {𝑥𝜒}))
6 df-sbc 3724 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
7 df-sbc 3724 . 2 ([𝐵 / 𝑥]𝜒𝐵 ∈ {𝑥𝜒})
85, 6, 73bitr4g 317 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  Ⅎwnf 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-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: (None)
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