Theorem csbcom2fi 35942

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 3801	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷}
2	1	abeq2i 2868	. . . 4 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷)
3		df-csb 3801	. . . . . 6 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐷}
4	3	abeq2i 2868	. . . . 5 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐷)
5	4	sbcbii 3745	. . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷)
6	2, 5	bitri 278	. . 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 3801	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐷}
11	10	abeq2i 2868	. . . . 5 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐷 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐷)
12		csbcom2fi.4	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸
13	12	eleq2i 2825	. . . . 5 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐷 ↔ 𝑧 ∈ 𝐸)
14	11, 13	bitr3i 280	. . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐸)
15	7, 8, 9, 14	sbccom2fi 35941	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷 ↔ [𝐶 / 𝑦]𝑧 ∈ 𝐸)
16		sbcel2 4315	. . 3 ⊢ ([𝐶 / 𝑦]𝑧 ∈ 𝐸 ↔ 𝑧 ∈ ⦋𝐶 / 𝑦⦌𝐸)
17	6, 15, 16	3bitri 300	. 2 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ 𝑧 ∈ ⦋𝐶 / 𝑦⦌𝐸)
18	17	eqriv 2736	1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸

Syntax hints:   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2880  Vcvv 3400  [wsbc 3685  ⦋csb 3800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2373  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-nul 4222

This theorem is referenced by: (None)