Theorem sbccom2f 39261

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

Hypotheses

Ref	Expression
sbccom2f.1	⊢ 𝐴 ∈ V
sbccom2f.2	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
sbccom2f	⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbccom2f

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbccow 3765	. . . 4 ⊢ ([𝐵 / 𝑧][𝑧 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑)
2	1	bicomi 227	. . 3 ⊢ ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
3	2	sbcbii 3798	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
4		sbccom2f.1	. . 3 ⊢ 𝐴 ∈ V
5	4	sbccom2 39260	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
6		vex 3458	. . . . . . 7 ⊢ 𝑧 ∈ V
7	6	sbccom2 39260	. . . . . 6 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [⦋𝑧 / 𝑦⦌𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
8		sbccom2f.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
9	6, 8	csbgfi 3871	. . . . . . 7 ⊢ ⦋𝑧 / 𝑦⦌𝐴 = 𝐴
10		dfsbcq 3744	. . . . . . 7 ⊢ (⦋𝑧 / 𝑦⦌𝐴 = 𝐴 → ([⦋𝑧 / 𝑦⦌𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑))
11	9, 10	ax-mp 5	. . . . . 6 ⊢ ([⦋𝑧 / 𝑦⦌𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
12	7, 11	bitri 278	. . . . 5 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
13	12	bicomi 227	. . . 4 ⊢ ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
14	13	sbcbii 3798	. . 3 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
15		sbccow 3765	. . 3 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
16	14, 15	bitri 278	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
17	3, 5, 16	3bitri 300	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911  Vcvv 3454  [wsbc 3742  ⦋csb 3851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456  df-sbc 3743  df-csb 3852

This theorem is used by:  sbccom2fi  39262