Theorem sbccom 3039

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 3038	. . . 4 ⊢ ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
2		sbccomlem 3038	. . . . . . 7 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
3	2	sbcbii 3023	. . . . . 6 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4		sbccomlem 3038	. . . . . 6 ⊢ ([𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
5	3, 4	bitri 184	. . . . 5 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
6	5	sbcbii 3023	. . . 4 ⊢ ([𝐴 / 𝑧][𝐵 / 𝑤][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
7		sbccomlem 3038	. . . . 5 ⊢ ([𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
8	7	sbcbii 3023	. . . 4 ⊢ ([𝐵 / 𝑤][𝐴 / 𝑧][𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
9	1, 6, 8	3bitr3i 210	. . 3 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
10		sbcco 2985	. . 3 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
11		sbcco 2985	. . 3 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
12	9, 10, 11	3bitr3i 210	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑)
13		sbcco 2985	. . 3 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑)
14	13	sbcbii 3023	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
15		sbcco 2985	. . 3 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
16	15	sbcbii 3023	. 2 ⊢ ([𝐵 / 𝑦][𝐴 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
17	12, 14, 16	3bitr3i 210	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 105  [wsbc 2963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-sbc 2964

This theorem is referenced by:  csbcomg  3081  csbabg  3119  mpoxopovel  6242