Theorem csbcomg 3025

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 2697	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 2697	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		sbccom 2984	. . . . . 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 3023	. . . . . . 7 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶))
6	5	sbcbidv 2967	. . . . . 6 ⊢ (𝐵 ∈ V → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶))
7	6	adantl 275	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶))
8		sbcel2g 3023	. . . . . . 7 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶))
9	8	sbcbidv 2967	. . . . . 6 ⊢ (𝐴 ∈ V → ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶))
10	9	adantr 274	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶))
11	4, 7, 10	3bitr3d 217	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶))
12		sbcel2g 3023	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶))
13	12	adantr 274	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶))
14		sbcel2g 3023	. . . . 5 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶))
15	14	adantl 275	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶))
16	11, 13, 15	3bitr3d 217	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶))
17	16	eqrdv 2137	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶)
18	1, 2, 17	syl2an 287	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  Vcvv 2686  [wsbc 2909  ⦋csb 3003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004

This theorem is referenced by:  ovmpos  5894