pncan2 - Intuitionistic Logic Explorer

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Theorem pncan2 8113

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 8043	. . . 4 $/\$
2	1	oveq1d 5865	. . 3 $/\$
3		pncan 8112	. . 3 $/\$
4	2, 3	eqtr3d 2205	. 2 $/\$
5	4	ancoms 266	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1348

wcel 2141 (class class class)co 5850

cc 7759

caddc 7764

cmin 8077

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7853 ax-1cn 7854 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-sub 8079

This theorem is referenced by: subid 8125 pnpcan 8145 pnncan 8147 pncan2d 8219 fzrev3 10030 fzrevral3 10050 fzosubel2 10138 facndiv 10660 bcnp1n 10680 trireciplem 11450