Theorem csbcomg 2850

Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)

Assertion

Ref	Expression
csbcomg	⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ⦋A / x⦌⦋B / y⦌𝐶 = ⦋B / y⦌⦋A / x⦌𝐶)

Distinct variable groups:   y,A   x,B   x,y

Allowed substitution hints:   A(x)   B(y)   𝐶(x,y)   𝑉(x,y)   𝑊(x,y)

Proof of Theorem csbcomg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2543	. 2 ⊢ (A ∈ 𝑉 → A ∈ V)
2		elex 2543	. 2 ⊢ (B ∈ 𝑊 → B ∈ V)
3		sbccom 2810	. . . . . 6 ⊢ ([A / x][B / y]z ∈ 𝐶 ↔ [B / y][A / x]z ∈ 𝐶)
4	3	a1i 9	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ([A / x][B / y]z ∈ 𝐶 ↔ [B / y][A / x]z ∈ 𝐶))
5		sbcel2g 2848	. . . . . . 7 ⊢ (B ∈ V → ([B / y]z ∈ 𝐶 ↔ z ∈ ⦋B / y⦌𝐶))
6	5	sbcbidv 2794	. . . . . 6 ⊢ (B ∈ V → ([A / x][B / y]z ∈ 𝐶 ↔ [A / x]z ∈ ⦋B / y⦌𝐶))
7	6	adantl 262	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ([A / x][B / y]z ∈ 𝐶 ↔ [A / x]z ∈ ⦋B / y⦌𝐶))
8		sbcel2g 2848	. . . . . . 7 ⊢ (A ∈ V → ([A / x]z ∈ 𝐶 ↔ z ∈ ⦋A / x⦌𝐶))
9	8	sbcbidv 2794	. . . . . 6 ⊢ (A ∈ V → ([B / y][A / x]z ∈ 𝐶 ↔ [B / y]z ∈ ⦋A / x⦌𝐶))
10	9	adantr 261	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ([B / y][A / x]z ∈ 𝐶 ↔ [B / y]z ∈ ⦋A / x⦌𝐶))
11	4, 7, 10	3bitr3d 207	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ([A / x]z ∈ ⦋B / y⦌𝐶 ↔ [B / y]z ∈ ⦋A / x⦌𝐶))
12		sbcel2g 2848	. . . . 5 ⊢ (A ∈ V → ([A / x]z ∈ ⦋B / y⦌𝐶 ↔ z ∈ ⦋A / x⦌⦋B / y⦌𝐶))
13	12	adantr 261	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ([A / x]z ∈ ⦋B / y⦌𝐶 ↔ z ∈ ⦋A / x⦌⦋B / y⦌𝐶))
14		sbcel2g 2848	. . . . 5 ⊢ (B ∈ V → ([B / y]z ∈ ⦋A / x⦌𝐶 ↔ z ∈ ⦋B / y⦌⦋A / x⦌𝐶))
15	14	adantl 262	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ([B / y]z ∈ ⦋A / x⦌𝐶 ↔ z ∈ ⦋B / y⦌⦋A / x⦌𝐶))
16	11, 13, 15	3bitr3d 207	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → (z ∈ ⦋A / x⦌⦋B / y⦌𝐶 ↔ z ∈ ⦋B / y⦌⦋A / x⦌𝐶))
17	16	eqrdv 2020	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → ⦋A / x⦌⦋B / y⦌𝐶 = ⦋B / y⦌⦋A / x⦌𝐶)
18	1, 2, 17	syl2an 273	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ⦋A / x⦌⦋B / y⦌𝐶 = ⦋B / y⦌⦋A / x⦌𝐶)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  Vcvv 2535  [wsbc 2741  ⦋csb 2829

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742  df-csb 2830

This theorem is referenced by:  ovmpt2s  5547