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Theorem csbcomg 2867
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 2560 . 2 (A 𝑉A V)
2 elex 2560 . 2 (B 𝑊B V)
3 sbccom 2827 . . . . . 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 2865 . . . . . . 7 (B V → ([B / y]z 𝐶z B / y𝐶))
65sbcbidv 2811 . . . . . 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 2865 . . . . . . 7 (A V → ([A / x]z 𝐶z A / x𝐶))
98sbcbidv 2811 . . . . . 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 2865 . . . . 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 2865 . . . . 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 2035 . 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 1242   wcel 1390  Vcvv 2551  [wsbc 2758  csb 2846
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by:  ovmpt2s  5566
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