Theorem sbccom2 36982

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		sbccow 3800	. . . . . . 7 ⊢ ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑)
2	1	bicomi 223	. . . . . 6 ⊢ ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
3	2	sbcbii 3837	. . . . 5 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
4		sbccow 3800	. . . . . 6 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
5	4	bicomi 223	. . . . 5 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
6		vex 3479	. . . . . . 7 ⊢ 𝑧 ∈ V
7	6	sbccom2lem 36981	. . . . . 6 ⊢ ([𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
8	7	sbcbii 3837	. . . . 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 297	. . . 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 36981	. . . 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		sbccow 3800	. . . . 5 ⊢ ([𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
13	12	sbcbii 3837	. . . 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 297	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
15		csbcow 3908	. . . 4 ⊢ ⦋𝐴 / 𝑧⦌⦋𝑧 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
16		dfsbcq 3779	. . . 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 275	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
19		sbccom 3865	. . 3 ⊢ ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝐴 / 𝑥]𝜑)
20	19	sbcbii 3837	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑥]𝜑)
21		sbccow 3800	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
22	18, 20, 21	3bitri 297	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2107  Vcvv 3475  [wsbc 3777  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477  df-sbc 3778  df-csb 3894

This theorem is referenced by:  sbccom2f  36983