Theorem caovcan 5629

Description: Convert an operation cancellation law to class notation. (Contributed by set.mm contributors, 20-Aug-1995.)

Hypotheses

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

Assertion

Ref	Expression
caovcan	⊢ ((A ∈ S ∧ B ∈ 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 caovcan

Step	Hyp	Ref	Expression
1		oveq1 5531	. . . 4 ⊢ (x = A → (xFy) = (AFy))
2		oveq1 5531	. . . 4 ⊢ (x = A → (xFC) = (AFC))
3	1, 2	eqeq12d 2367	. . 3 ⊢ (x = A → ((xFy) = (xFC) ↔ (AFy) = (AFC)))
4	3	imbi1d 308	. 2 ⊢ (x = A → (((xFy) = (xFC) → y = C) ↔ ((AFy) = (AFC) → y = C)))
5		oveq2 5532	. . . 4 ⊢ (y = B → (AFy) = (AFB))
6	5	eqeq1d 2361	. . 3 ⊢ (y = B → ((AFy) = (AFC) ↔ (AFB) = (AFC)))
7		eqeq1 2359	. . 3 ⊢ (y = B → (y = C ↔ B = C))
8	6, 7	imbi12d 311	. 2 ⊢ (y = B → (((AFy) = (AFC) → y = C) ↔ ((AFB) = (AFC) → B = C)))
9		caovcan.1	. . 3 ⊢ C ∈ V
10		oveq2 5532	. . . . . 6 ⊢ (z = C → (xFz) = (xFC))
11	10	eqeq2d 2364	. . . . 5 ⊢ (z = C → ((xFy) = (xFz) ↔ (xFy) = (xFC)))
12		eqeq2 2362	. . . . 5 ⊢ (z = C → (y = z ↔ y = C))
13	11, 12	imbi12d 311	. . . 4 ⊢ (z = C → (((xFy) = (xFz) → y = z) ↔ ((xFy) = (xFC) → y = C)))
14	13	imbi2d 307	. . 3 ⊢ (z = C → (((x ∈ S ∧ y ∈ S) → ((xFy) = (xFz) → y = z)) ↔ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFC) → y = C))))
15		caovcan.2	. . 3 ⊢ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFz) → y = z))
16	9, 14, 15	vtocl 2910	. 2 ⊢ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFC) → y = C))
17	4, 8, 16	vtocl2ga 2923	1 ⊢ ((A ∈ S ∧ B ∈ S) → ((AFB) = (AFC) → B = C))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860  (class class class)co 5526

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  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  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-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796  df-ov 5527

This theorem is referenced by: (None)