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Theorem csbcomg 3025
 Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
csbcomg ((𝐴𝑉𝐵𝑊) → 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 2697 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 2697 . 2 (𝐵𝑊𝐵 ∈ V)
3 sbccom 2984 . . . . . 6 ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶)
43a1i 9 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶))
5 sbcel2g 3023 . . . . . . 7 (𝐵 ∈ V → ([𝐵 / 𝑦]𝑧𝐶𝑧𝐵 / 𝑦𝐶))
65sbcbidv 2967 . . . . . 6 (𝐵 ∈ V → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶))
76adantl 275 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶))
8 sbcel2g 3023 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧𝐶𝑧𝐴 / 𝑥𝐶))
98sbcbidv 2967 . . . . . 6 (𝐴 ∈ V → ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶))
109adantr 274 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶))
114, 7, 103bitr3d 217 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶))
12 sbcel2g 3023 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶))
1312adantr 274 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶))
14 sbcel2g 3023 . . . . 5 (𝐵 ∈ V → ([𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶))
1514adantl 275 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶))
1611, 13, 153bitr3d 217 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶))
1716eqrdv 2137 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
181, 2, 17syl2an 287 1 ((𝐴𝑉𝐵𝑊) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  Vcvv 2686  [wsbc 2909  ⦋csb 3003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004 This theorem is referenced by:  ovmpos  5894
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