Theorem sbccom 2889

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 2888	. . . 4 ⊢ ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
2		sbccomlem 2888	. . . . . . 7 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
3	2	sbcbii 2873	. . . . . 6 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4		sbccomlem 2888	. . . . . 6 ⊢ ([𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
5	3, 4	bitri 182	. . . . 5 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
6	5	sbcbii 2873	. . . 4 ⊢ ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
7		sbccomlem 2888	. . . . 5 ⊢ ([𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
8	7	sbcbii 2873	. . . 4 ⊢ ([𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
9	1, 6, 8	3bitr3i 208	. . 3 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
10		sbcco 2836	. . 3 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
11		sbcco 2836	. . 3 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
12	9, 10, 11	3bitr3i 208	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
13		sbcco 2836	. . 3 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑)
14	13	sbcbii 2873	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
15		sbcco 2836	. . 3 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
16	15	sbcbii 2873	. 2 ⊢ ([𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
17	12, 14, 16	3bitr3i 208	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 103  [wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  csbcomg  2929  csbabg  2963  mpt2xopovel  5879