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Theorem sbcimi 34221
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 3675 . . 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 341 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜒𝜂))
73, 6bitri 266 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wcel 2156  Vcvv 3391  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-v 3393  df-sbc 3634
This theorem is referenced by: (None)
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