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Theorem sbcimi 36277
Description: Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
Hypotheses
Ref Expression
sbcimi.1 𝐴 ∈ V
sbcimi.2 ([𝐴 / 𝑥]𝜑𝜒)
sbcimi.3 ([𝐴 / 𝑥]𝜓𝜂)
Assertion
Ref Expression
sbcimi ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))

Proof of Theorem sbcimi
StepHypRef Expression
1 sbcimi.1 . . 3 𝐴 ∈ V
2 sbcimg 3771 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
31, 2ax-mp 5 . 2 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
4 sbcimi.2 . . 3 ([𝐴 / 𝑥]𝜑𝜒)
5 sbcimi.3 . . 3 ([𝐴 / 𝑥]𝜓𝜂)
64, 5imbi12i 351 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜒𝜂))
73, 6bitri 274 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2110  Vcvv 3431  [wsbc 3720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-sbc 3721
This theorem is referenced by: (None)
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