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Theorem csbcomg 3159
 Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
csbcomg ((A V B W) → [A / x][B / y]C = [B / y][A / x]C)
Distinct variable groups:   y,A   x,B   x,y
Allowed substitution hints:   A(x)   B(y)   C(x,y)   V(x,y)   W(x,y)

Proof of Theorem csbcomg
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2 (A VA V)
2 elex 2867 . 2 (B WB V)
3 sbccom 3117 . . . . . 6 ([̣A / x]̣[̣B / yz C ↔ [̣B / y]̣[̣A / xz C)
43a1i 10 . . . . 5 ((A V B V) → ([̣A / x]̣[̣B / yz C ↔ [̣B / y]̣[̣A / xz C))
5 sbcel2g 3157 . . . . . . 7 (B V → ([̣B / yz Cz [B / y]C))
65sbcbidv 3100 . . . . . 6 (B V → ([̣A / x]̣[̣B / yz C ↔ [̣A / xz [B / y]C))
76adantl 452 . . . . 5 ((A V B V) → ([̣A / x]̣[̣B / yz C ↔ [̣A / xz [B / y]C))
8 sbcel2g 3157 . . . . . . 7 (A V → ([̣A / xz Cz [A / x]C))
98sbcbidv 3100 . . . . . 6 (A V → ([̣B / y]̣[̣A / xz C ↔ [̣B / yz [A / x]C))
109adantr 451 . . . . 5 ((A V B V) → ([̣B / y]̣[̣A / xz C ↔ [̣B / yz [A / x]C))
114, 7, 103bitr3d 274 . . . 4 ((A V B V) → ([̣A / xz [B / y]C ↔ [̣B / yz [A / x]C))
12 sbcel2g 3157 . . . . 5 (A V → ([̣A / xz [B / y]Cz [A / x][B / y]C))
1312adantr 451 . . . 4 ((A V B V) → ([̣A / xz [B / y]Cz [A / x][B / y]C))
14 sbcel2g 3157 . . . . 5 (B V → ([̣B / yz [A / x]Cz [B / y][A / x]C))
1514adantl 452 . . . 4 ((A V B V) → ([̣B / yz [A / x]Cz [B / y][A / x]C))
1611, 13, 153bitr3d 274 . . 3 ((A V B V) → (z [A / x][B / y]Cz [B / y][A / x]C))
1716eqrdv 2351 . 2 ((A V B V) → [A / x][B / y]C = [B / y][A / x]C)
181, 2, 17syl2an 463 1 ((A V B W) → [A / x][B / y]C = [B / y][A / x]C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [̣wsbc 3046  [csb 3136 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-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 This theorem is referenced by: (None)
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