Theorem csbcom2fi 38135

Description: Commutative law for double class substitution in a class, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)

Hypotheses

Ref	Expression
csbcom2fi.1	⊢ 𝐴 ∈ V
csbcom2fi.2	⊢ Ⅎ𝑦𝐴
csbcom2fi.3	⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
csbcom2fi.4	⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸

Assertion

Ref	Expression
csbcom2fi	⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)

Proof of Theorem csbcom2fi

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3900	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷}
2	1	eqabri 2885	. . . 4 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷)
3		df-csb 3900	. . . . . 6 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐷}
4	3	eqabri 2885	. . . . 5 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐷)
5	4	sbcbii 3846	. . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷)
6	2, 5	bitri 275	. . 3 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷)
7		csbcom2fi.1	. . . 4 ⊢ 𝐴 ∈ V
8		csbcom2fi.2	. . . 4 ⊢ Ⅎ𝑦𝐴
9		csbcom2fi.3	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
10		df-csb 3900	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐷}
11	10	eqabri 2885	. . . . 5 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐷 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐷)
12		csbcom2fi.4	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸
13	12	eleq2i 2833	. . . . 5 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐷 ↔ 𝑧 ∈ 𝐸)
14	11, 13	bitr3i 277	. . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐸)
15	7, 8, 9, 14	sbccom2fi 38134	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷 ↔ [𝐶 / 𝑦]𝑧 ∈ 𝐸)
16		sbcel2 4418	. . 3 ⊢ ([𝐶 / 𝑦]𝑧 ∈ 𝐸 ↔ 𝑧 ∈ ⦋𝐶 / 𝑦⦌𝐸)
17	6, 15, 16	3bitri 297	. 2 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ 𝑧 ∈ ⦋𝐶 / 𝑦⦌𝐸)
18	17	eqriv 2734	1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸

Syntax hints:   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  Vcvv 3480  [wsbc 3788  ⦋csb 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-nul 4334

This theorem is referenced by: (None)