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Theorem sbcimdv 3028
Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1457). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
Hypothesis
Ref Expression
sbcimdv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcimdv (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem sbcimdv
StepHypRef Expression
1 sbcex 2971 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
2 sbcimdv.1 . . . . 5 (𝜑 → (𝜓𝜒))
32alrimiv 1874 . . . 4 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbc 2974 . . . 4 (𝐴 ∈ V → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
5 sbcim1 3011 . . . 4 ([𝐴 / 𝑥](𝜓𝜒) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
63, 4, 5syl56 34 . . 3 (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
76com3l 81 . 2 (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒)))
81, 7mpdi 43 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351  wcel 2148  Vcvv 2737  [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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  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-nfc 2308  df-v 2739  df-sbc 2963
This theorem is referenced by: (None)
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