Theorem riotaeqimp 5985

Description: If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)

Hypotheses

Ref	Expression
riotaeqimp.i	|- I = ( iota_ a e. V X = A )
riotaeqimp.j	|- J = ( iota_ a e. V Y = A )
riotaeqimp.x	|- ( ph -> E! a e. V X = A )
riotaeqimp.y	|- ( ph -> E! a e. V Y = A )

Assertion

Ref	Expression
riotaeqimp	|- ( ( ph /\ I = J ) -> X = Y )

Distinct variable groups:   I, a   J, a   V, a   X, a   Y, a

Allowed substitution hints:   ph( a)   A( a)

Proof of Theorem riotaeqimp

Step	Hyp	Ref	Expression
1		riotaeqimp.j	. . . . . . 7 |- J = ( iota_ a e. V Y = A )
2	1	eqcomi 2233	. . . . . 6 |- ( iota_ a e. V Y = A ) = J
3	2	eqeq2i 2240	. . . . 5 |- ( I = ( iota_ a e. V Y = A ) <-> I = J )
4	3	a1i 9	. . . 4 |- ( ph -> ( I = ( iota_ a e. V Y = A ) <-> I = J ) )
5	4	bicomd 141	. . 3 |- ( ph -> ( I = J <-> I = ( iota_ a e. V Y = A ) ) )
6	5	biimpa 296	. 2 |- ( ( ph /\ I = J ) -> I = ( iota_ a e. V Y = A ) )
7		riotaeqimp.i	. . . . 5 |- I = ( iota_ a e. V X = A )
8	7	eqeq1i 2237	. . . 4 |- ( I = J <-> ( iota_ a e. V X = A ) = J )
9		riotaeqimp.y	. . . . . . 7 |- ( ph -> E! a e. V Y = A )
10		riotacl 5976	. . . . . . 7 |- ( E! a e. V Y = A -> ( iota_ a e. V Y = A ) e. V )
11	9, 10	syl 14	. . . . . 6 |- ( ph -> ( iota_ a e. V Y = A ) e. V )
12	1, 11	eqeltrid 2316	. . . . 5 |- ( ph -> J e. V )
13		riotaeqimp.x	. . . . 5 |- ( ph -> E! a e. V X = A )
14		nfv 1574	. . . . . . 7 |- F/ a J e. V
15		nfcvd 2373	. . . . . . 7 |- ( J e. V -> F/_ a J )
16		nfcvd 2373	. . . . . . . 8 |- ( J e. V -> F/_ a X )
17	15	nfcsb1d 3155	. . . . . . . 8 |- ( J e. V -> F/_ a [_ J / a ]_ A )
18	16, 17	nfeqd 2387	. . . . . . 7 |- ( J e. V -> F/ a X = [_ J / a ]_ A )
19		id 19	. . . . . . 7 |- ( J e. V -> J e. V )
20		csbeq1a 3133	. . . . . . . . 9 |- ( a = J -> A = [_ J / a ]_ A )
21	20	eqeq2d 2241	. . . . . . . 8 |- ( a = J -> ( X = A <-> X = [_ J / a ]_ A ) )
22	21	adantl 277	. . . . . . 7 |- ( ( J e. V /\ a = J ) -> ( X = A <-> X = [_ J / a ]_ A ) )
23	14, 15, 18, 19, 22	riota2df 5982	. . . . . 6 |- ( ( J e. V /\ E! a e. V X = A ) -> ( X = [_ J / a ]_ A <-> ( iota_ a e. V X = A ) = J ) )
24	23	bicomd 141	. . . . 5 |- ( ( J e. V /\ E! a e. V X = A ) -> ( ( iota_ a e. V X = A ) = J <-> X = [_ J / a ]_ A ) )
25	12, 13, 24	syl2anc 411	. . . 4 |- ( ph -> ( ( iota_ a e. V X = A ) = J <-> X = [_ J / a ]_ A ) )
26	8, 25	bitrid 192	. . 3 |- ( ph -> ( I = J <-> X = [_ J / a ]_ A ) )
27	26	biimpa 296	. 2 |- ( ( ph /\ I = J ) -> X = [_ J / a ]_ A )
28		riotacl 5976	. . . . . . . 8 |- ( E! a e. V X = A -> ( iota_ a e. V X = A ) e. V )
29	13, 28	syl 14	. . . . . . 7 |- ( ph -> ( iota_ a e. V X = A ) e. V )
30	7, 29	eqeltrid 2316	. . . . . 6 |- ( ph -> I e. V )
31		nfv 1574	. . . . . . 7 |- F/ a I e. V
32		nfcvd 2373	. . . . . . 7 |- ( I e. V -> F/_ a I )
33		nfcvd 2373	. . . . . . . 8 |- ( I e. V -> F/_ a Y )
34	32	nfcsb1d 3155	. . . . . . . 8 |- ( I e. V -> F/_ a [_ I / a ]_ A )
35	33, 34	nfeqd 2387	. . . . . . 7 |- ( I e. V -> F/ a Y = [_ I / a ]_ A )
36		id 19	. . . . . . 7 |- ( I e. V -> I e. V )
37		csbeq1a 3133	. . . . . . . . 9 |- ( a = I -> A = [_ I / a ]_ A )
38	37	eqeq2d 2241	. . . . . . . 8 |- ( a = I -> ( Y = A <-> Y = [_ I / a ]_ A ) )
39	38	adantl 277	. . . . . . 7 |- ( ( I e. V /\ a = I ) -> ( Y = A <-> Y = [_ I / a ]_ A ) )
40	31, 32, 35, 36, 39	riota2df 5982	. . . . . 6 |- ( ( I e. V /\ E! a e. V Y = A ) -> ( Y = [_ I / a ]_ A <-> ( iota_ a e. V Y = A ) = I ) )
41	30, 9, 40	syl2anc 411	. . . . 5 |- ( ph -> ( Y = [_ I / a ]_ A <-> ( iota_ a e. V Y = A ) = I ) )
42		eqcom 2231	. . . . 5 |- ( ( iota_ a e. V Y = A ) = I <-> I = ( iota_ a e. V Y = A ) )
43	41, 42	bitrdi 196	. . . 4 |- ( ph -> ( Y = [_ I / a ]_ A <-> I = ( iota_ a e. V Y = A ) ) )
44	43	adantr 276	. . 3 |- ( ( ph /\ I = J ) -> ( Y = [_ I / a ]_ A <-> I = ( iota_ a e. V Y = A ) ) )
45		csbeq1 3127	. . . . . . 7 |- ( J = I -> [_ J / a ]_ A = [_ I / a ]_ A )
46	45	eqcoms 2232	. . . . . 6 |- ( I = J -> [_ J / a ]_ A = [_ I / a ]_ A )
47		eqeq12 2242	. . . . . . 7 |- ( ( X = [_ J / a ]_ A /\ Y = [_ I / a ]_ A ) -> ( X = Y <-> [_ J / a ]_ A = [_ I / a ]_ A ) )
48	47	ancoms 268	. . . . . 6 |- ( ( Y = [_ I / a ]_ A /\ X = [_ J / a ]_ A ) -> ( X = Y <-> [_ J / a ]_ A = [_ I / a ]_ A ) )
49	46, 48	syl5ibrcom 157	. . . . 5 |- ( I = J -> ( ( Y = [_ I / a ]_ A /\ X = [_ J / a ]_ A ) -> X = Y ) )
50	49	expd 258	. . . 4 |- ( I = J -> ( Y = [_ I / a ]_ A -> ( X = [_ J / a ]_ A -> X = Y ) ) )
51	50	adantl 277	. . 3 |- ( ( ph /\ I = J ) -> ( Y = [_ I / a ]_ A -> ( X = [_ J / a ]_ A -> X = Y ) ) )
52	44, 51	sylbird 170	. 2 |- ( ( ph /\ I = J ) -> ( I = ( iota_ a e. V Y = A ) -> ( X = [_ J / a ]_ A -> X = Y ) ) )
53	6, 27, 52	mp2d 47	1 |- ( ( ph /\ I = J ) -> X = Y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E!wreu 2510   [_csb 3124   iota_crio 5959

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-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-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-uni 3889  df-iota 5278  df-riota 5960

This theorem is referenced by:  uspgredg2v  16027