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Theorem csbcomg 2850
Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
csbcomg ((A 𝑉 B 𝑊) → A / xB / y𝐶 = B / yA / x𝐶)
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 2543 . 2 (A 𝑉A V)
2 elex 2543 . 2 (B 𝑊B V)
3 sbccom 2810 . . . . . 6 ([A / x][B / y]z 𝐶[B / y][A / x]z 𝐶)
43a1i 9 . . . . 5 ((A V B V) → ([A / x][B / y]z 𝐶[B / y][A / x]z 𝐶))
5 sbcel2g 2848 . . . . . . 7 (B V → ([B / y]z 𝐶z B / y𝐶))
65sbcbidv 2794 . . . . . 6 (B V → ([A / x][B / y]z 𝐶[A / x]z B / y𝐶))
76adantl 262 . . . . 5 ((A V B V) → ([A / x][B / y]z 𝐶[A / x]z B / y𝐶))
8 sbcel2g 2848 . . . . . . 7 (A V → ([A / x]z 𝐶z A / x𝐶))
98sbcbidv 2794 . . . . . 6 (A V → ([B / y][A / x]z 𝐶[B / y]z A / x𝐶))
109adantr 261 . . . . 5 ((A V B V) → ([B / y][A / x]z 𝐶[B / y]z A / x𝐶))
114, 7, 103bitr3d 207 . . . 4 ((A V B V) → ([A / x]z B / y𝐶[B / y]z A / x𝐶))
12 sbcel2g 2848 . . . . 5 (A V → ([A / x]z B / y𝐶z A / xB / y𝐶))
1312adantr 261 . . . 4 ((A V B V) → ([A / x]z B / y𝐶z A / xB / y𝐶))
14 sbcel2g 2848 . . . . 5 (B V → ([B / y]z A / x𝐶z B / yA / x𝐶))
1514adantl 262 . . . 4 ((A V B V) → ([B / y]z A / x𝐶z B / yA / x𝐶))
1611, 13, 153bitr3d 207 . . 3 ((A V B V) → (z A / xB / y𝐶z B / yA / x𝐶))
1716eqrdv 2020 . 2 ((A V B V) → A / xB / y𝐶 = B / yA / x𝐶)
181, 2, 17syl2an 273 1 ((A 𝑉 B 𝑊) → A / xB / y𝐶 = B / yA / x𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  [wsbc 2741  csb 2829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742  df-csb 2830
This theorem is referenced by:  ovmpt2s  5547
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