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Theorem sbcimi 35380
 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 3818 . . 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 353 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜒𝜂))
73, 6bitri 277 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2107  Vcvv 3493  [wsbc 3770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-sbc 3771 This theorem is referenced by: (None)
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