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Theorem sbccom 3871
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 3869 . . . 4 ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
2 sbccomlem 3869 . . . . . . 7 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
32sbcbii 3846 . . . . . 6 ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4 sbccomlem 3869 . . . . . 6 ([𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
53, 4bitri 275 . . . . 5 ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
65sbcbii 3846 . . . 4 ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
7 sbccomlem 3869 . . . . 5 ([𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
87sbcbii 3846 . . . 4 ([𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
91, 6, 83bitr3i 301 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
10 sbccow 3811 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
11 sbccow 3811 . . 3 ([𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
129, 10, 113bitr3i 301 . 2 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
13 sbccow 3811 . . 3 ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
1413sbcbii 3846 . 2 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
15 sbccow 3811 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
1615sbcbii 3846 . 2 ([𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
1712, 14, 163bitr3i 301 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sbc 3789
This theorem is referenced by:  csbcom  4420  csbab  4440  mpoxopovel  8245  fi1uzind  14546  wrd2ind  14761  elmptrab  23835  rspc2daf  32485  sbccom2  38132  sbcrot3  42802  ichnreuop  47459  ichreuopeq  47460
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