pncan2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > pncan2		Unicode version

Theorem pncan2 8130

Description: Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)

**Assertion**
Ref	Expression
pncan2	$/\$

**Proof of Theorem pncan2**
Step	Hyp	Ref	Expression
1		addcom 8060	. . . 4 $/\$
2	1	oveq1d 5872	. . 3 $/\$
3		pncan 8129	. . 3 $/\$
4	2, 3	eqtr3d 2206	. 2 $/\$
5	4	ancoms 266	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1349

wcel 2142 (class class class)co 5857

cc 7776

caddc 7781

cmin 8094

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 610 ax-in2 611 ax-io 705 ax-5 1441 ax-7 1442 ax-gen 1443 ax-ie1 1487 ax-ie2 1488 ax-8 1498 ax-10 1499 ax-11 1500 ax-i12 1501 ax-bndl 1503 ax-4 1504 ax-17 1520 ax-i9 1524 ax-ial 1528 ax-i5r 1529 ax-14 2145 ax-ext 2153 ax-sep 4108 ax-pow 4161 ax-pr 4195 ax-setind 4522 ax-resscn 7870 ax-1cn 7871 ax-icn 7873 ax-addcl 7874 ax-addrcl 7875 ax-mulcl 7876 ax-addcom 7878 ax-addass 7880 ax-distr 7882 ax-i2m1 7883 ax-0id 7886 ax-rnegex 7887 ax-cnre 7889

This theorem depends on definitions: df-bi 116 df-3an 976 df-tru 1352 df-fal 1355 df-nf 1455 df-sb 1757 df-eu 2023 df-mo 2024 df-clab 2158 df-cleq 2164 df-clel 2167 df-nfc 2302 df-ne 2342 df-ral 2454 df-rex 2455 df-reu 2456 df-rab 2458 df-v 2733 df-sbc 2957 df-dif 3124 df-un 3126 df-in 3128 df-ss 3135 df-pw 3569 df-sn 3590 df-pr 3591 df-op 3593 df-uni 3798 df-br 3991 df-opab 4052 df-id 4279 df-xp 4618 df-rel 4619 df-cnv 4620 df-co 4621 df-dm 4622 df-iota 5162 df-fun 5202 df-fv 5208 df-riota 5813 df-ov 5860 df-oprab 5861 df-mpo 5862 df-sub 8096

This theorem is referenced by: subid 8142 pnpcan 8162 pnncan 8164 pncan2d 8236 fzrev3 10047 fzrevral3 10067 fzosubel2 10155 facndiv 10677 bcnp1n 10697 trireciplem 11467