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Theorem csbresg 4683
Description: Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbresg (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem csbresg
StepHypRef Expression
1 csbing 3196 . . 3 (𝐴𝑉𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)))
2 csbxpg 4486 . . . . 5 (𝐴𝑉𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V))
3 csbconstg 2934 . . . . . 6 (𝐴𝑉𝐴 / 𝑥V = V)
43xpeq2d 4434 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
52, 4eqtrd 2117 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
65ineq2d 3190 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
71, 6eqtrd 2117 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
8 df-res 4422 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
98csbeq2i 2946 . 2 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))
10 df-res 4422 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
117, 9, 103eqtr4g 2142 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  Vcvv 2615  csb 2922  cin 2987   × cxp 4408  cres 4412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-in 2994  df-opab 3875  df-xp 4416  df-res 4422
This theorem is referenced by: (None)
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