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Theorem sbccom 3117
 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 / yφ ↔ [̣B / y]̣[̣A / xφ)
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 3116 . . . 4 ([̣A / z]̣[̣B / w]̣[̣w / y]̣[̣z / xφ ↔ [̣B / w]̣[̣A / z]̣[̣w / y]̣[̣z / xφ)
2 sbccomlem 3116 . . . . . . 7 ([̣w / y]̣[̣z / xφ ↔ [̣z / x]̣[̣w / yφ)
32sbcbii 3101 . . . . . 6 ([̣B / w]̣[̣w / y]̣[̣z / xφ ↔ [̣B / w]̣[̣z / x]̣[̣w / yφ)
4 sbccomlem 3116 . . . . . 6 ([̣B / w]̣[̣z / x]̣[̣w / yφ ↔ [̣z / x]̣[̣B / w]̣[̣w / yφ)
53, 4bitri 240 . . . . 5 ([̣B / w]̣[̣w / y]̣[̣z / xφ ↔ [̣z / x]̣[̣B / w]̣[̣w / yφ)
65sbcbii 3101 . . . 4 ([̣A / z]̣[̣B / w]̣[̣w / y]̣[̣z / xφ ↔ [̣A / z]̣[̣z / x]̣[̣B / w]̣[̣w / yφ)
7 sbccomlem 3116 . . . . 5 ([̣A / z]̣[̣w / y]̣[̣z / xφ ↔ [̣w / y]̣[̣A / z]̣[̣z / xφ)
87sbcbii 3101 . . . 4 ([̣B / w]̣[̣A / z]̣[̣w / y]̣[̣z / xφ ↔ [̣B / w]̣[̣w / y]̣[̣A / z]̣[̣z / xφ)
91, 6, 83bitr3i 266 . . 3 ([̣A / z]̣[̣z / x]̣[̣B / w]̣[̣w / yφ ↔ [̣B / w]̣[̣w / y]̣[̣A / z]̣[̣z / xφ)
10 sbcco 3068 . . 3 ([̣A / z]̣[̣z / x]̣[̣B / w]̣[̣w / yφ ↔ [̣A / x]̣[̣B / w]̣[̣w / yφ)
11 sbcco 3068 . . 3 ([̣B / w]̣[̣w / y]̣[̣A / z]̣[̣z / xφ ↔ [̣B / y]̣[̣A / z]̣[̣z / xφ)
129, 10, 113bitr3i 266 . 2 ([̣A / x]̣[̣B / w]̣[̣w / yφ ↔ [̣B / y]̣[̣A / z]̣[̣z / xφ)
13 sbcco 3068 . . 3 ([̣B / w]̣[̣w / yφ ↔ [̣B / yφ)
1413sbcbii 3101 . 2 ([̣A / x]̣[̣B / w]̣[̣w / yφ ↔ [̣A / x]̣[̣B / yφ)
15 sbcco 3068 . . 3 ([̣A / z]̣[̣z / xφ ↔ [̣A / xφ)
1615sbcbii 3101 . 2 ([̣B / y]̣[̣A / z]̣[̣z / xφ ↔ [̣B / y]̣[̣A / xφ)
1712, 14, 163bitr3i 266 1 ([̣A / x]̣[̣B / yφ ↔ [̣B / y]̣[̣A / xφ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  csbcomg  3159  csbabg  3197  cnvopab  5030  eqerlem  5960
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