Theorem sbccom2 33100

Description: Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)

Hypothesis

Ref	Expression
sbccom2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
sbccom2	⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem sbccom2

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcco 3425	. . . . . . 7 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑)
2	1	bicomi 213	. . . . . 6 ⊢ ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
3	2	sbcbii 3458	. . . . 5 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
4		sbcco 3425	. . . . . 6 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
5	4	bicomi 213	. . . . 5 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
6		vex 3176	. . . . . . 7 ⊢ 𝑧 ∈ V
7	6	sbccom2lem 33099	. . . . . 6 ⊢ ([𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
8	7	sbcbii 3458	. . . . 5 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑧][⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9	3, 5, 8	3bitri 285	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑧][⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10		sbccom2.1	. . . . 5 ⊢ 𝐴 ∈ V
11	10	sbccom2lem 33099	. . . 4 ⊢ ([𝐴 / 𝑧][⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
12		sbcco 3425	. . . . 5 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
13	12	sbcbii 3458	. . . 4 ⊢ ([⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
14	9, 11, 13	3bitri 285	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
15		csbco 3509	. . . 4 ⊢ ⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
16		dfsbcq 3404	. . . 4 ⊢ (⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 → ([⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑))
17	15, 16	ax-mp 5	. . 3 ⊢ ([⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
18	14, 17	bitri 263	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
19		sbccom 3476	. . 3 ⊢ ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝐴 / 𝑥]𝜑)
20	19	sbcbii 3458	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑥]𝜑)
21		sbcco 3425	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
22	18, 20, 21	3bitri 285	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500

This theorem is referenced by:  sbccom2f  33101