Theorem csbcomg 3160

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 2868	. 2 ⊢ (A ∈ V → A ∈ V)
2		elex 2868	. 2 ⊢ (B ∈ W → B ∈ V)
3		sbccom 3118	. . . . . 6 ⊢ ([̣A / x]̣[̣B / y]̣z ∈ C ↔ [̣B / y]̣[̣A / x]̣z ∈ C)
4	3	a1i 10	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ([̣A / x]̣[̣B / y]̣z ∈ C ↔ [̣B / y]̣[̣A / x]̣z ∈ C))
5		sbcel2g 3158	. . . . . . 7 ⊢ (B ∈ V → ([̣B / y]̣z ∈ C ↔ z ∈ [B / y]C))
6	5	sbcbidv 3101	. . . . . 6 ⊢ (B ∈ V → ([̣A / x]̣[̣B / y]̣z ∈ C ↔ [̣A / x]̣z ∈ [B / y]C))
7	6	adantl 452	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ([̣A / x]̣[̣B / y]̣z ∈ C ↔ [̣A / x]̣z ∈ [B / y]C))
8		sbcel2g 3158	. . . . . . 7 ⊢ (A ∈ V → ([̣A / x]̣z ∈ C ↔ z ∈ [A / x]C))
9	8	sbcbidv 3101	. . . . . 6 ⊢ (A ∈ V → ([̣B / y]̣[̣A / x]̣z ∈ C ↔ [̣B / y]̣z ∈ [A / x]C))
10	9	adantr 451	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ([̣B / y]̣[̣A / x]̣z ∈ C ↔ [̣B / y]̣z ∈ [A / x]C))
11	4, 7, 10	3bitr3d 274	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ([̣A / x]̣z ∈ [B / y]C ↔ [̣B / y]̣z ∈ [A / x]C))
12		sbcel2g 3158	. . . . 5 ⊢ (A ∈ V → ([̣A / x]̣z ∈ [B / y]C ↔ z ∈ [A / x][B / y]C))
13	12	adantr 451	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ([̣A / x]̣z ∈ [B / y]C ↔ z ∈ [A / x][B / y]C))
14		sbcel2g 3158	. . . . 5 ⊢ (B ∈ V → ([̣B / y]̣z ∈ [A / x]C ↔ z ∈ [B / y][A / x]C))
15	14	adantl 452	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ([̣B / y]̣z ∈ [A / x]C ↔ z ∈ [B / y][A / x]C))
16	11, 13, 15	3bitr3d 274	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → (z ∈ [A / x][B / y]C ↔ z ∈ [B / y][A / x]C))
17	16	eqrdv 2351	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → [A / x][B / y]C = [B / y][A / x]C)
18	1, 2, 17	syl2an 463	1 ⊢ ((A ∈ V ∧ B ∈ W) → [A / x][B / y]C = [B / y][A / x]C)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [̣wsbc 3047  [csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)