Theorem subcan 8327

Description: Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
subcan	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = ( A - C ) <-> B = C ) )

Proof of Theorem subcan

Step	Hyp	Ref	Expression
1		simp2 1001	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
2		simp1 1000	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
3	1, 2	addcomd 8223	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + A ) = ( A + B ) )
4	3	eqeq1d 2214	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) = ( A + C ) <-> ( A + B ) = ( A + C ) ) )
5		simp3 1002	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
6		addsubeq4 8287	. . 3 |- ( ( ( B e. CC /\ A e. CC ) /\ ( A e. CC /\ C e. CC ) ) -> ( ( B + A ) = ( A + C ) <-> ( A - B ) = ( A - C ) ) )
7	1, 2, 2, 5, 6	syl22anc 1251	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) = ( A + C ) <-> ( A - B ) = ( A - C ) ) )
8		addcan 8252	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
9	4, 7, 8	3bitr3d 218	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = ( A - C ) <-> B = C ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176  (class class class)co 5944   CCcc 7923   + caddc 7928   - cmin 8243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585  ax-resscn 8017  ax-1cn 8018  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-sub 8245

This theorem is referenced by:  subcani  8365  subcand  8424  subcanad  8426