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Theorem sbccom 3856
 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 3855 . . . 4 ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
2 sbccomlem 3855 . . . . . . 7 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
32sbcbii 3831 . . . . . 6 ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4 sbccomlem 3855 . . . . . 6 ([𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
53, 4bitri 277 . . . . 5 ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
65sbcbii 3831 . . . 4 ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
7 sbccomlem 3855 . . . . 5 ([𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
87sbcbii 3831 . . . 4 ([𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑[𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
91, 6, 83bitr3i 303 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
10 sbccow 3797 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
11 sbccow 3797 . . 3 ([𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
129, 10, 113bitr3i 303 . 2 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
13 sbccow 3797 . . 3 ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
1413sbcbii 3831 . 2 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
15 sbccow 3797 . . 3 ([𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
1615sbcbii 3831 . 2 ([𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
1712, 14, 163bitr3i 303 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208  [wsbc 3774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-v 3498  df-sbc 3775 This theorem is referenced by:  csbcom  4371  csbab  4391  mpoxopovel  7888  fi1uzind  13858  wrd2ind  14087  elmptrab  22437  rspc2daf  30233  sbccom2  35405  sbcrot3  39395  ichnreuop  43641  ichreuopeq  43642
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