Theorem cdaeng 6074

Description: Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by Paul Chapman, 5-Jun-2009.)

Assertion

Ref	Expression
cdaeng	|- (((A e. W /\ B e. X) /\ (C e. Y /\ D e. Z) /\ (A ~~ B /\ C ~~ D)) -> (A +c C) ~~ (B +c D))

Proof of Theorem cdaeng

Step	Hyp	Ref	Expression
1		breq1 3341	. . . . . . 7 |- (z = C -> (z ~~ w <-> C ~~ w))
2	1	anbi2d 678	. . . . . 6 |- (z = C -> ((A ~~ B /\ z ~~ w) <-> (A ~~ B /\ C ~~ w)))
3		opreq2 4890	. . . . . . 7 |- (z = C -> (A +c z) = (A +c C))
4	3	breq1d 3348	. . . . . 6 |- (z = C -> ((A +c z) ~~ (B +c w) <-> (A +c C) ~~ (B +c w)))
5	2, 4	imbi12d 688	. . . . 5 |- (z = C -> (((A ~~ B /\ z ~~ w) -> (A +c z) ~~ (B +c w)) <-> ((A ~~ B /\ C ~~ w) -> (A +c C) ~~ (B +c w))))
6	5	imbi2d 674	. . . 4 |- (z = C -> (((A e. W /\ B e. X) -> ((A ~~ B /\ z ~~ w) -> (A +c z) ~~ (B +c w))) <-> ((A e. W /\ B e. X) -> ((A ~~ B /\ C ~~ w) -> (A +c C) ~~ (B +c w)))))
7		breq2 3342	. . . . . . 7 |- (w = D -> (C ~~ w <-> C ~~ D))
8	7	anbi2d 678	. . . . . 6 |- (w = D -> ((A ~~ B /\ C ~~ w) <-> (A ~~ B /\ C ~~ D)))
9		opreq2 4890	. . . . . . 7 |- (w = D -> (B +c w) = (B +c D))
10	9	breq2d 3350	. . . . . 6 |- (w = D -> ((A +c C) ~~ (B +c w) <-> (A +c C) ~~ (B +c D)))
11	8, 10	imbi12d 688	. . . . 5 |- (w = D -> (((A ~~ B /\ C ~~ w) -> (A +c C) ~~ (B +c w)) <-> ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))))
12	11	imbi2d 674	. . . 4 |- (w = D -> (((A e. W /\ B e. X) -> ((A ~~ B /\ C ~~ w) -> (A +c C) ~~ (B +c w))) <-> ((A e. W /\ B e. X) -> ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D)))))
13		breq1 3341	. . . . . . 7 |- (x = A -> (x ~~ y <-> A ~~ y))
14	13	anbi1d 679	. . . . . 6 |- (x = A -> ((x ~~ y /\ z ~~ w) <-> (A ~~ y /\ z ~~ w)))
15		opreq1 4889	. . . . . . 7 |- (x = A -> (x +c z) = (A +c z))
16	15	breq1d 3348	. . . . . 6 |- (x = A -> ((x +c z) ~~ (y +c w) <-> (A +c z) ~~ (y +c w)))
17	14, 16	imbi12d 688	. . . . 5 |- (x = A -> (((x ~~ y /\ z ~~ w) -> (x +c z) ~~ (y +c w)) <-> ((A ~~ y /\ z ~~ w) -> (A +c z) ~~ (y +c w))))
18		breq2 3342	. . . . . . 7 |- (y = B -> (A ~~ y <-> A ~~ B))
19	18	anbi1d 679	. . . . . 6 |- (y = B -> ((A ~~ y /\ z ~~ w) <-> (A ~~ B /\ z ~~ w)))
20		opreq1 4889	. . . . . . 7 |- (y = B -> (y +c w) = (B +c w))
21	20	breq2d 3350	. . . . . 6 |- (y = B -> ((A +c z) ~~ (y +c w) <-> (A +c z) ~~ (B +c w)))
22	19, 21	imbi12d 688	. . . . 5 |- (y = B -> (((A ~~ y /\ z ~~ w) -> (A +c z) ~~ (y +c w)) <-> ((A ~~ B /\ z ~~ w) -> (A +c z) ~~ (B +c w))))
23		visset 2295	. . . . . 6 |- x e. _V
24		visset 2295	. . . . . 6 |- y e. _V
25		visset 2295	. . . . . 6 |- z e. _V
26		visset 2295	. . . . . 6 |- w e. _V
27	23, 24, 25, 26	cdaen 6073	. . . . 5 |- ((x ~~ y /\ z ~~ w) -> (x +c z) ~~ (y +c w))
28	17, 22, 27	vtocl2g 2349	. . . 4 |- ((A e. W /\ B e. X) -> ((A ~~ B /\ z ~~ w) -> (A +c z) ~~ (B +c w)))
29	6, 12, 28	vtocl2g 2349	. . 3 |- ((C e. Y /\ D e. Z) -> ((A e. W /\ B e. X) -> ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))))
30	29	impcom 378	. 2 |- (((A e. W /\ B e. X) /\ (C e. Y /\ D e. Z)) -> ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D)))
31	30	3impia 1064	1 |- (((A e. W /\ B e. X) /\ (C e. Y /\ D e. Z) /\ (A ~~ B /\ C ~~ D)) -> (A +c C) ~~ (B +c D))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  (class class class)co 4884   ~~ cen 5423   +c ccda 6065

This theorem is referenced by:  nnaun 6089

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1o 5177  df-er 5318  df-en 5427  df-cda 6066

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