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Theorem csbcomg 2954
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 2630 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 2630 . 2 (𝐵𝑊𝐵 ∈ V)
3 sbccom 2914 . . . . . 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 2952 . . . . . . 7 (𝐵 ∈ V → ([𝐵 / 𝑦]𝑧𝐶𝑧𝐵 / 𝑦𝐶))
65sbcbidv 2897 . . . . . 6 (𝐵 ∈ V → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶))
76adantl 271 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶))
8 sbcel2g 2952 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧𝐶𝑧𝐴 / 𝑥𝐶))
98sbcbidv 2897 . . . . . 6 (𝐴 ∈ V → ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶))
109adantr 270 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶))
114, 7, 103bitr3d 216 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶))
12 sbcel2g 2952 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶))
1312adantr 270 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶))
14 sbcel2g 2952 . . . . 5 (𝐵 ∈ V → ([𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶))
1514adantl 271 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶))
1611, 13, 153bitr3d 216 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶))
1716eqrdv 2086 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
181, 2, 17syl2an 283 1 ((𝐴𝑉𝐵𝑊) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  Vcvv 2619  [wsbc 2840  csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  ovmpt2s  5760
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