wkslem1 - Intuitionistic Logic Explorer

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Theorem wkslem1 16026

Description: Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)

**Assertion**
Ref	Expression
wkslem1	if- if-

**Proof of Theorem wkslem1**
Step	Hyp	Ref	Expression
1		fveq2 5626	. . 3
2		fvoveq1 6023	. . 3
3	1, 2	eqeq12d 2244	. 2
4		2fveq3 5631	. . 3
5	1	sneqd 3679	. . 3
6	4, 5	eqeq12d 2244	. 2
7	1, 2	preq12d 3751	. . 3
8	7, 4	sseq12d 3255	. 2
9	3, 6, 8	ifpbi123d 998	1 if- if-

Colors of variables: wff set class

Syntax hints:

wi 4

wb 105 if-wif 983

wceq 1395

wss 3197

csn 3666

cpr 3667

cfv 5317 (class class class)co 6000

c1 7996

caddc 7998

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-ext 2211

This theorem depends on definitions: df-bi 117 df-ifp 984 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-rex 2514 df-v 2801 df-un 3201 df-in 3203 df-ss 3210 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-br 4083 df-iota 5277 df-fv 5325 df-ov 6003

This theorem is referenced by: (None)