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Theorem sbcimi 34042
 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 3510 . . 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 339 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜒𝜂))
73, 6bitri 264 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2030  Vcvv 3231  [wsbc 3468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-sbc 3469 This theorem is referenced by: (None)
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