Theorem sbccom2f 35406

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		sbcco 3800	. . . 4 ⊢ ([𝐵 / 𝑧][𝑧 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑)
2	1	bicomi 226	. . 3 ⊢ ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
3	2	sbcbii 3831	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
4		sbccom2f.1	. . 3 ⊢ 𝐴 ∈ V
5	4	sbccom2 35405	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
6		vex 3499	. . . . . . 7 ⊢ 𝑧 ∈ V
7	6	sbccom2 35405	. . . . . 6 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [⦋𝑧 / 𝑦⦌𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
8		sbccom2f.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
9	6, 8	csbgfi 3905	. . . . . . 7 ⊢ ⦋𝑧 / 𝑦⦌𝐴 = 𝐴
10		dfsbcq 3776	. . . . . . 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 277	. . . . 5 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
13	12	bicomi 226	. . . 4 ⊢ ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
14	13	sbcbii 3831	. . 3 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
15		sbcco 3800	. . 3 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
16	14, 15	bitri 277	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
17	3, 5, 16	3bitri 299	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 208   = wceq 1537   ∈ wcel 2114  Ⅎwnfc 2963  Vcvv 3496  [wsbc 3774  ⦋csb 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-sbc 3775  df-csb 3886

This theorem is referenced by:  sbccom2fi  35407