Theorem caoprcan 4055

Description: Convert an operation cancellation law to class notation.

Hypotheses

Ref	Expression
caoprcan.1	|- C e. V
caoprcan.2	|- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))

Assertion

Ref	Expression
caoprcan	|- ((A e. S /\ B e. S) -> ((AFB) = (AFC) -> B = C))

Distinct variable groups:   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 3968	. . . 4 |- (x = A -> (xFy) = (AFy))
2		opreq1 3968	. . . 4 |- (x = A -> (xFC) = (AFC))
3	1, 2	eqeq12d 1489	. . 3 |- (x = A -> ((xFy) = (xFC) <-> (AFy) = (AFC)))
4	3	imbi1d 613	. 2 |- (x = A -> (((xFy) = (xFC) -> y = C) <-> ((AFy) = (AFC) -> y = C)))
5		opreq2 3969	. . . 4 |- (y = B -> (AFy) = (AFB))
6	5	eqeq1d 1483	. . 3 |- (y = B -> ((AFy) = (AFC) <-> (AFB) = (AFC)))
7		eqeq1 1481	. . 3 |- (y = B -> (y = C <-> B = C))
8	6, 7	imbi12d 626	. 2 |- (y = B -> (((AFy) = (AFC) -> y = C) <-> ((AFB) = (AFC) -> B = C)))
9		caoprcan.1	. . 3 |- C e. V
10		opreq2 3969	. . . . . 6 |- (z = C -> (xFz) = (xFC))
11	10	eqeq2d 1486	. . . . 5 |- (z = C -> ((xFy) = (xFz) <-> (xFy) = (xFC)))
12		eqeq2 1484	. . . . 5 |- (z = C -> (y = z <-> y = C))
13	11, 12	imbi12d 626	. . . 4 |- (z = C -> (((xFy) = (xFz) -> y = z) <-> ((xFy) = (xFC) -> y = C)))
14	13	imbi2d 612	. . 3 |- (z = C -> (((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z)) <-> ((x e. S /\ y e. S) -> ((xFy) = (xFC) -> y = C))))
15		caoprcan.2	. . 3 |- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))
16	9, 14, 15	vtocl 1842	. 2 |- ((x e. S /\ y e. S) -> ((xFy) = (xFC) -> y = C))
17	4, 8, 16	vtocl2ga 1853	1 |- ((A e. S /\ B e. S) -> ((AFB) = (AFC) -> B = C))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  (class class class)co 3963

This theorem is referenced by:  ecopoprtrn 4311

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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