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Theorem sbcimdvOLD 3485
Description: Obsolete proof of sbcimdv 3484 as of 7-Jul-2021. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
sbcimdv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcimdvOLD (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem sbcimdvOLD
StepHypRef Expression
1 sbcimdv.1 . . . 4 (𝜑 → (𝜓𝜒))
21alrimiv 1852 . . 3 (𝜑 → ∀𝑥(𝜓𝜒))
3 spsbc 3434 . . 3 (𝐴 ∈ V → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
4 sbcim1 3468 . . 3 ([𝐴 / 𝑥](𝜓𝜒) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
52, 3, 4syl56 36 . 2 (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
6 sbcex 3431 . . . . 5 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
76con3i 150 . . . 4 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜓)
87pm2.21d 118 . . 3 𝐴 ∈ V → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
98a1d 25 . 2 𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
105, 9pm2.61i 176 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478  wcel 1987  Vcvv 3189  [wsbc 3421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3191  df-sbc 3422
This theorem is referenced by: (None)
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