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.

Step	Hyp	Ref	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)
3	2	sbcbii 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)
5	3, 4	bitri 275	. . . . 5 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
6	5	sbcbii 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)
8	7	sbcbii 3846	. . . 4 ⊢ ([𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
9	1, 6, 8	3bitr3i 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)
12	9, 10, 11	3bitr3i 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)
14	13	sbcbii 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)
16	15	sbcbii 3846	. 2 ⊢ ([𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
17	12, 14, 16	3bitr3i 301	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

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