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Theorem csbresg 4976
 Description: Distribute proper substitution through the restriction of a class. csbresg 4976 is derived from the virtual deduction proof csbresgVD in set.mm. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbresg (A V[A / x](B C) = ([A / x]B [A / x]C))

Proof of Theorem csbresg
StepHypRef Expression
1 csbing 3462 . . 3 (A V[A / x](B ∩ (C × V)) = ([A / x]B[A / x](C × V)))
2 csbxpg 4813 . . . . 5 (A V[A / x](C × V) = ([A / x]C × [A / x]V))
3 csbconstg 3150 . . . . . 6 (A V[A / x]V = V)
43xpeq2d 4808 . . . . 5 (A V → ([A / x]C × [A / x]V) = ([A / x]C × V))
52, 4eqtrd 2385 . . . 4 (A V[A / x](C × V) = ([A / x]C × V))
65ineq2d 3457 . . 3 (A V → ([A / x]B[A / x](C × V)) = ([A / x]B ∩ ([A / x]C × V)))
71, 6eqtrd 2385 . 2 (A V[A / x](B ∩ (C × V)) = ([A / x]B ∩ ([A / x]C × V)))
8 df-res 4788 . . 3 (B C) = (B ∩ (C × V))
98csbeq2i 3162 . 2 [A / x](B C) = [A / x](B ∩ (C × V))
10 df-res 4788 . 2 ([A / x]B [A / x]C) = ([A / x]B ∩ ([A / x]C × V))
117, 9, 103eqtr4g 2410 1 (A V[A / x](B C) = ([A / x]B [A / x]C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [csb 3136   ∩ cin 3208   × cxp 4770   ↾ cres 4774 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-3an 936  df-nan 1288  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  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-opab 4623  df-xp 4784  df-res 4788 This theorem is referenced by: (None)
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