Theorem caoprcan 3069

Description: Convert an operation cancellation law to class notation.

Hypotheses

Ref	Expression
caoprcan.1	⊢ C ∈ V
caoprcan.2	⊢ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFz) → y = z))

Assertion

Ref	Expression
caoprcan	⊢ ((A ∈ S ∧ B ∈ S) → ((AFB) = (AFC) → B = C))

Distinct variable group(s):   x,y,z,F   x,S,y,z   x,A,y,z   x,B,y,z   x,C,y,z

Proof of Theorem caoprcan

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . 4 ⊢ (x = A → (xFy) = (AFy))
2		opreq1 3006	. . . 4 ⊢ (x = A → (xFC) = (AFC))
3	1, 2	cleq12d 1115	. . 3 ⊢ (x = A → ((xFy) = (xFC) ↔ (AFy) = (AFC)))
4	3	imbi1d 465	. 2 ⊢ (x = A → (((xFy) = (xFC) → y = C) ↔ ((AFy) = (AFC) → y = C)))
5		opreq2 3007	. . . 4 ⊢ (y = B → (AFy) = (AFB))
6	5	cleq1d 1109	. . 3 ⊢ (y = B → ((AFy) = (AFC) ↔ (AFB) = (AFC)))
7		cleq1 1107	. . 3 ⊢ (y = B → (y = C ↔ B = C))
8	6, 7	imbi12d 474	. 2 ⊢ (y = B → (((AFy) = (AFC) → y = C) ↔ ((AFB) = (AFC) → B = C)))
9		caoprcan.1	. . 3 ⊢ C ∈ V
10		opreq2 3007	. . . . . 6 ⊢ (z = C → (xFz) = (xFC))
11	10	cleq2d 1112	. . . . 5 ⊢ (z = C → ((xFy) = (xFz) ↔ (xFy) = (xFC)))
12		cleq2 1110	. . . . 5 ⊢ (z = C → (y = z ↔ y = C))
13	11, 12	imbi12d 474	. . . 4 ⊢ (z = C → (((xFy) = (xFz) → y = z) ↔ ((xFy) = (xFC) → y = C)))
14	13	imbi2d 464	. . 3 ⊢ (z = C → (((x ∈ S ∧ y ∈ S) → ((xFy) = (xFz) → y = z)) ↔ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFC) → y = C))))
15		caoprcan.2	. . 3 ⊢ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFz) → y = z))
16	9, 14, 15	vtocl 1378	. 2 ⊢ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFC) → y = C))
17	4, 8, 16	vtocl2ga 1388	1 ⊢ ((A ∈ S ∧ B ∈ S) → ((AFB) = (AFC) → B = C))

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348  (class class class)co 3001

This theorem is referenced by:  ecopoprtrn 3247

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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