Theorem frds 5935

Description: Substitution schema verson of frd 5922. (Contributed by SF, 19-Mar-2015.)

Hypotheses

Ref	Expression
frds.1	|- {x | ps} e. _V
frds.2	|- (x = y -> (ps <-> ch))
frds.3	|- (x = z -> (ps <-> th))
frds.4	|- (ph -> R Fr A)
frds.5	|- (ph -> E.x e. A ps)

Assertion

Ref	Expression
frds	|- (ph -> E.y e. A (ch /\ A.z e. A ((th /\ zRy) -> z = y)))

Distinct variable groups:   x,A,y,z   ch,x   ps,y,z   y,R,z   th,x   x,y,z

Allowed substitution hints:   ph(x,y,z)   ps(x)   ch(y,z)   th(y,z)   R(x)

Proof of Theorem frds

Step	Hyp	Ref	Expression
1		frds.4	. . 3 |- (ph -> R Fr A)
2		dfrab2 3530	. . . . 5 |- {x e. A | ps} = ({x | ps} i^i A)
3		df-rab 2623	. . . . 5 |- {x e. A | ps} = {x | (x e. A /\ ps)}
4	2, 3	eqtr3i 2375	. . . 4 |- ({x | ps} i^i A) = {x | (x e. A /\ ps)}
5		frds.1	. . . . 5 |- {x | ps} e. _V
6		brex 4689	. . . . . . 7 |- (R Fr A -> (R e. _V /\ A e. _V))
7	1, 6	syl 15	. . . . . 6 |- (ph -> (R e. _V /\ A e. _V))
8	7	simprd 449	. . . . 5 |- (ph -> A e. _V)
9		inexg 4100	. . . . 5 |- (({x | ps} e. _V /\ A e. _V) -> ({x | ps} i^i A) e. _V)
10	5, 8, 9	sylancr 644	. . . 4 |- (ph -> ({x | ps} i^i A) e. _V)
11	4, 10	syl5eqelr 2438	. . 3 |- (ph -> {x | (x e. A /\ ps)} e. _V)
12		ssab2 3350	. . . 4 |- {x | (x e. A /\ ps)} C_ A
13	12	a1i 10	. . 3 |- (ph -> {x | (x e. A /\ ps)} C_ A)
14		frds.5	. . . . 5 |- (ph -> E.x e. A ps)
15		df-rex 2620	. . . . 5 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
16	14, 15	sylib 188	. . . 4 |- (ph -> E.x(x e. A /\ ps))
17		abn0 3568	. . . 4 |- ({x | (x e. A /\ ps)} =/= (/) <-> E.x(x e. A /\ ps))
18	16, 17	sylibr 203	. . 3 |- (ph -> {x | (x e. A /\ ps)} =/= (/))
19	1, 11, 13, 18	frd 5922	. 2 |- (ph -> E.y e. {x | (x e. A /\ ps)}A.z e. {x | (x e. A /\ ps)} (zRy -> z = y))
20		eleq1 2413	. . . . . 6 |- (x = y -> (x e. A <-> y e. A))
21		frds.2	. . . . . 6 |- (x = y -> (ps <-> ch))
22	20, 21	anbi12d 691	. . . . 5 |- (x = y -> ((x e. A /\ ps) <-> (y e. A /\ ch)))
23	22	rexab 2999	. . . 4 |- (E.y e. {x | (x e. A /\ ps)}A.z e. A ((th /\ zRy) -> z = y) <-> E.y((y e. A /\ ch) /\ A.z e. A ((th /\ zRy) -> z = y)))
24		anass 630	. . . . 5 |- (((y e. A /\ ch) /\ A.z e. A ((th /\ zRy) -> z = y)) <-> (y e. A /\ (ch /\ A.z e. A ((th /\ zRy) -> z = y))))
25	24	exbii 1582	. . . 4 |- (E.y((y e. A /\ ch) /\ A.z e. A ((th /\ zRy) -> z = y)) <-> E.y(y e. A /\ (ch /\ A.z e. A ((th /\ zRy) -> z = y))))
26	23, 25	bitri 240	. . 3 |- (E.y e. {x | (x e. A /\ ps)}A.z e. A ((th /\ zRy) -> z = y) <-> E.y(y e. A /\ (ch /\ A.z e. A ((th /\ zRy) -> z = y))))
27		impexp 433	. . . . . . 7 |- (((z e. A /\ th) -> (zRy -> z = y)) <-> (z e. A -> (th -> (zRy -> z = y))))
28		impexp 433	. . . . . . . 8 |- (((th /\ zRy) -> z = y) <-> (th -> (zRy -> z = y)))
29	28	imbi2i 303	. . . . . . 7 |- ((z e. A -> ((th /\ zRy) -> z = y)) <-> (z e. A -> (th -> (zRy -> z = y))))
30	27, 29	bitr4i 243	. . . . . 6 |- (((z e. A /\ th) -> (zRy -> z = y)) <-> (z e. A -> ((th /\ zRy) -> z = y)))
31	30	albii 1566	. . . . 5 |- (A.z((z e. A /\ th) -> (zRy -> z = y)) <-> A.z(z e. A -> ((th /\ zRy) -> z = y)))
32		eleq1 2413	. . . . . . 7 |- (x = z -> (x e. A <-> z e. A))
33		frds.3	. . . . . . 7 |- (x = z -> (ps <-> th))
34	32, 33	anbi12d 691	. . . . . 6 |- (x = z -> ((x e. A /\ ps) <-> (z e. A /\ th)))
35	34	ralab 2997	. . . . 5 |- (A.z e. {x | (x e. A /\ ps)} (zRy -> z = y) <-> A.z((z e. A /\ th) -> (zRy -> z = y)))
36		df-ral 2619	. . . . 5 |- (A.z e. A ((th /\ zRy) -> z = y) <-> A.z(z e. A -> ((th /\ zRy) -> z = y)))
37	31, 35, 36	3bitr4i 268	. . . 4 |- (A.z e. {x | (x e. A /\ ps)} (zRy -> z = y) <-> A.z e. A ((th /\ zRy) -> z = y))
38	37	rexbii 2639	. . 3 |- (E.y e. {x | (x e. A /\ ps)}A.z e. {x | (x e. A /\ ps)} (zRy -> z = y) <-> E.y e. {x | (x e. A /\ ps)}A.z e. A ((th /\ zRy) -> z = y))
39		df-rex 2620	. . 3 |- (E.y e. A (ch /\ A.z e. A ((th /\ zRy) -> z = y)) <-> E.y(y e. A /\ (ch /\ A.z e. A ((th /\ zRy) -> z = y))))
40	26, 38, 39	3bitr4i 268	. 2 |- (E.y e. {x | (x e. A /\ ps)}A.z e. {x | (x e. A /\ ps)} (zRy -> z = y) <-> E.y e. A (ch /\ A.z e. A ((th /\ zRy) -> z = y)))
41	19, 40	sylib 188	1 |- (ph -> E.y e. A (ch /\ A.z e. A ((th /\ zRy) -> z = y)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   e. wcel 1710  {cab 2339   =/= wne 2516  A.wral 2614  E.wrex 2615  {crab 2618  _Vcvv 2859   i^i cin 3208   C_ wss 3257  (/)c0 3550   class class class wbr 4639   Fr cfound 5894

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-found 5905

This theorem is referenced by:  weds  5938