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Theorem sbcbid 3020
Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
sbcbid.1 𝑥𝜑
sbcbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcbid (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Proof of Theorem sbcbid
StepHypRef Expression
1 sbcbid.1 . . . 4 𝑥𝜑
2 sbcbid.2 . . . 4 (𝜑 → (𝜓𝜒))
31, 2abbid 2294 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
43eleq2d 2247 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐴 ∈ {𝑥𝜒}))
5 df-sbc 2963 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 2963 . 2 ([𝐴 / 𝑥]𝜒𝐴 ∈ {𝑥𝜒})
74, 5, 63bitr4g 223 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wnf 1460  wcel 2148  {cab 2163  [wsbc 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2963
This theorem is referenced by:  sbcbidv  3021  csbeq2d  3082  bezoutlemstep  11968
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