Theorem caoprcan 4047

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 3959	. . . 4 |- (x = A -> (xFy) = (AFy))
2		opreq1 3959	. . . 4 |- (x = A -> (xFC) = (AFC))
3	1, 2	eqeq12d 1486	. . 3 |- (x = A -> ((xFy) = (xFC) <-> (AFy) = (AFC)))
4	3	imbi1d 612	. 2 |- (x = A -> (((xFy) = (xFC) -> y = C) <-> ((AFy) = (AFC) -> y = C)))
5		opreq2 3960	. . . 4 |- (y = B -> (AFy) = (AFB))
6	5	eqeq1d 1480	. . 3 |- (y = B -> ((AFy) = (AFC) <-> (AFB) = (AFC)))
7		eqeq1 1478	. . 3 |- (y = B -> (y = C <-> B = C))
8	6, 7	imbi12d 625	. 2 |- (y = B -> (((AFy) = (AFC) -> y = C) <-> ((AFB) = (AFC) -> B = C)))
9		caoprcan.1	. . 3 |- C e. V
10		opreq2 3960	. . . . . 6 |- (z = C -> (xFz) = (xFC))
11	10	eqeq2d 1483	. . . . 5 |- (z = C -> ((xFy) = (xFz) <-> (xFy) = (xFC)))
12		eqeq2 1481	. . . . 5 |- (z = C -> (y = z <-> y = C))
13	11, 12	imbi12d 625	. . . 4 |- (z = C -> (((xFy) = (xFz) -> y = z) <-> ((xFy) = (xFC) -> y = C)))
14	13	imbi2d 611	. . 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 1838	. 2 |- ((x e. S /\ y e. S) -> ((xFy) = (xFC) -> y = C))
17	4, 8, 16	vtocl2ga 1849	1 |- ((A e. S /\ B e. S) -> ((AFB) = (AFC) -> B = C))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807  (class class class)co 3954

This theorem is referenced by:  ecopoprtrn 4301

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956

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