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 / 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.

Step	Hyp	Ref	Expression
1		sbccomlem 3116	. . . 4 ⊢ ([̣A / z]̣[̣B / w]̣[̣w / y]̣[̣z / x]̣φ ↔ [̣B / w]̣[̣A / z]̣[̣w / y]̣[̣z / x]̣φ)
2		sbccomlem 3116	. . . . . . 7 ⊢ ([̣w / y]̣[̣z / x]̣φ ↔ [̣z / x]̣[̣w / y]̣φ)
3	2	sbcbii 3101	. . . . . 6 ⊢ ([̣B / w]̣[̣w / y]̣[̣z / x]̣φ ↔ [̣B / w]̣[̣z / x]̣[̣w / y]̣φ)
4		sbccomlem 3116	. . . . . 6 ⊢ ([̣B / w]̣[̣z / x]̣[̣w / y]̣φ ↔ [̣z / x]̣[̣B / w]̣[̣w / y]̣φ)
5	3, 4	bitri 240	. . . . 5 ⊢ ([̣B / w]̣[̣w / y]̣[̣z / x]̣φ ↔ [̣z / x]̣[̣B / w]̣[̣w / y]̣φ)
6	5	sbcbii 3101	. . . 4 ⊢ ([̣A / z]̣[̣B / w]̣[̣w / y]̣[̣z / x]̣φ ↔ [̣A / z]̣[̣z / x]̣[̣B / w]̣[̣w / y]̣φ)
7		sbccomlem 3116	. . . . 5 ⊢ ([̣A / z]̣[̣w / y]̣[̣z / x]̣φ ↔ [̣w / y]̣[̣A / z]̣[̣z / x]̣φ)
8	7	sbcbii 3101	. . . 4 ⊢ ([̣B / w]̣[̣A / z]̣[̣w / y]̣[̣z / x]̣φ ↔ [̣B / w]̣[̣w / y]̣[̣A / z]̣[̣z / x]̣φ)
9	1, 6, 8	3bitr3i 266	. . 3 ⊢ ([̣A / z]̣[̣z / x]̣[̣B / w]̣[̣w / y]̣φ ↔ [̣B / w]̣[̣w / y]̣[̣A / z]̣[̣z / x]̣φ)
10		sbcco 3068	. . 3 ⊢ ([̣A / z]̣[̣z / x]̣[̣B / w]̣[̣w / y]̣φ ↔ [̣A / x]̣[̣B / w]̣[̣w / y]̣φ)
11		sbcco 3068	. . 3 ⊢ ([̣B / w]̣[̣w / y]̣[̣A / z]̣[̣z / x]̣φ ↔ [̣B / y]̣[̣A / z]̣[̣z / x]̣φ)
12	9, 10, 11	3bitr3i 266	. 2 ⊢ ([̣A / x]̣[̣B / w]̣[̣w / y]̣φ ↔ [̣B / y]̣[̣A / z]̣[̣z / x]̣φ)
13		sbcco 3068	. . 3 ⊢ ([̣B / w]̣[̣w / y]̣φ ↔ [̣B / y]̣φ)
14	13	sbcbii 3101	. 2 ⊢ ([̣A / x]̣[̣B / w]̣[̣w / y]̣φ ↔ [̣A / x]̣[̣B / y]̣φ)
15		sbcco 3068	. . 3 ⊢ ([̣A / z]̣[̣z / x]̣φ ↔ [̣A / x]̣φ)
16	15	sbcbii 3101	. 2 ⊢ ([̣B / y]̣[̣A / z]̣[̣z / x]̣φ ↔ [̣B / y]̣[̣A / x]̣φ)
17	12, 14, 16	3bitr3i 266	1 ⊢ ([̣A / x]̣[̣B / y]̣φ ↔ [̣B / y]̣[̣A / x]̣φ)

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