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Theorem sbccom 3542
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 3541 . . . 4 ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
2 sbccomlem 3541 . . . . . . 7 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
32sbcbii 3524 . . . . . 6 ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4 sbccomlem 3541 . . . . . 6 ([𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
53, 4bitri 264 . . . . 5 ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
65sbcbii 3524 . . . 4 ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
7 sbccomlem 3541 . . . . 5 ([𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
87sbcbii 3524 . . . 4 ([𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
91, 6, 83bitr3i 290 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
10 sbcco 3491 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
11 sbcco 3491 . . 3 ([𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
129, 10, 113bitr3i 290 . 2 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
13 sbcco 3491 . . 3 ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
1413sbcbii 3524 . 2 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
15 sbcco 3491 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
1615sbcbii 3524 . 2 ([𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
1712, 14, 163bitr3i 290 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-sbc 3469
This theorem is referenced by:  csbcom  4027  csbab  4041  mpt2xopovel  7391  fi1uzind  13317  wrd2ind  13523  elmptrab  21678  sbccom2  34060  sbcrot3  37672  csbabgOLD  39367
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