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Theorem sbcimg 3087
 Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
sbcimg (A V → ([̣A / x]̣(φψ) ↔ ([̣A / xφ → [̣A / xψ)))

Proof of Theorem sbcimg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . 2 (y = A → ([y / x](φψ) ↔ [̣A / x]̣(φψ)))
2 dfsbcq2 3049 . . 3 (y = A → ([y / x]φ ↔ [̣A / xφ))
3 dfsbcq2 3049 . . 3 (y = A → ([y / x]ψ ↔ [̣A / xψ))
42, 3imbi12d 311 . 2 (y = A → (([y / x]φ → [y / x]ψ) ↔ ([̣A / xφ → [̣A / xψ)))
5 sbim 2065 . 2 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
61, 4, 5vtoclbg 2915 1 (A V → ([̣A / x]̣(φψ) ↔ ([̣A / xφ → [̣A / xψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  sbceqal  3097  sbcimdv  3107  sbc19.21g  3110  sbcss  3660  iota4an  4358
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