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Theorem sbcimg 2798
Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
sbcimg (A 𝑉 → ([A / x](φψ) ↔ ([A / x]φ[A / x]ψ)))

Proof of Theorem sbcimg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . 2 (y = A → ([y / x](φψ) ↔ [A / x](φψ)))
2 dfsbcq2 2761 . . 3 (y = A → ([y / x]φ[A / x]φ))
3 dfsbcq2 2761 . . 3 (y = A → ([y / x]ψ[A / x]ψ))
42, 3imbi12d 223 . 2 (y = A → (([y / x]φ → [y / x]ψ) ↔ ([A / x]φ[A / x]ψ)))
5 sbim 1824 . 2 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
61, 4, 5vtoclbg 2608 1 (A 𝑉 → ([A / x](φψ) ↔ ([A / x]φ[A / x]ψ)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  [wsb 1642  [wsbc 2758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  sbceqal  2808  sbcimdv  2817  sbc19.21g  2820  sbcssg  3324  iota4an  4829  sbcfung  4868  riotass2  5437
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