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Theorem sbccom 2890
 Description: Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
Assertion
Ref Expression
sbccom ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem sbccom
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbccomlem 2889 . . . 4 ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
2 sbccomlem 2889 . . . . . . 7 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
32sbcbii 2874 . . . . . 6 ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4 sbccomlem 2889 . . . . . 6 ([𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
53, 4bitri 182 . . . . 5 ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
65sbcbii 2874 . . . 4 ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
7 sbccomlem 2889 . . . . 5 ([𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
87sbcbii 2874 . . . 4 ([𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
91, 6, 83bitr3i 208 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
10 sbcco 2837 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
11 sbcco 2837 . . 3 ([𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
129, 10, 113bitr3i 208 . 2 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
13 sbcco 2837 . . 3 ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
1413sbcbii 2874 . 2 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
15 sbcco 2837 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
1615sbcbii 2874 . 2 ([𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
1712, 14, 163bitr3i 208 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 103  [wsbc 2816 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817 This theorem is referenced by:  csbcomg  2930  csbabg  2964  mpt2xopovel  5890
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