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Theorem csbresg 4558
Description: Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbresg (A 𝑉A / x(B𝐶) = (A / xBA / x𝐶))

Proof of Theorem csbresg
StepHypRef Expression
1 csbing 3138 . . 3 (A 𝑉A / x(B ∩ (𝐶 × V)) = (A / xBA / x(𝐶 × V)))
2 csbxpg 4364 . . . . 5 (A 𝑉A / x(𝐶 × V) = (A / x𝐶 × A / xV))
3 csbconstg 2858 . . . . . 6 (A 𝑉A / xV = V)
43xpeq2d 4312 . . . . 5 (A 𝑉 → (A / x𝐶 × A / xV) = (A / x𝐶 × V))
52, 4eqtrd 2069 . . . 4 (A 𝑉A / x(𝐶 × V) = (A / x𝐶 × V))
65ineq2d 3132 . . 3 (A 𝑉 → (A / xBA / x(𝐶 × V)) = (A / xB ∩ (A / x𝐶 × V)))
71, 6eqtrd 2069 . 2 (A 𝑉A / x(B ∩ (𝐶 × V)) = (A / xB ∩ (A / x𝐶 × V)))
8 df-res 4300 . . 3 (B𝐶) = (B ∩ (𝐶 × V))
98csbeq2i 2870 . 2 A / x(B𝐶) = A / x(B ∩ (𝐶 × V))
10 df-res 4300 . 2 (A / xBA / x𝐶) = (A / xB ∩ (A / x𝐶 × V))
117, 9, 103eqtr4g 2094 1 (A 𝑉A / x(B𝐶) = (A / xBA / x𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  csb 2846  cin 2910   × cxp 4286  cres 4290
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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-in 2918  df-opab 3810  df-xp 4294  df-res 4300
This theorem is referenced by: (None)
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