Theorem acexmidlemv 6073

Description: Lemma for acexmid 6074.

This is acexmid 6074 with additional disjoint variable conditions, most notably between ph and x.

(Contributed by Jim Kingdon, 6-Aug-2019.)

Hypothesis

Ref	Expression
acexmidlemv.choice	|- E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u )

Assertion

Ref	Expression
acexmidlemv	|- ( ph \/ -. ph )

Distinct variable group:   ph, x, y, z, w, v, u

Proof of Theorem acexmidlemv

Dummy variables s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsucelsucexmidlem 4671	. . . 4 |- { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } e. On
2		pp0ex 4321	. . . . 5 |- { (/) , { (/) } } e. _V
3	2	rabex 4275	. . . 4 |- { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } e. _V
4		prexg 4344	. . . 4 |- ( ( { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } e. On /\ { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } e. _V ) -> { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } e. _V )
5	1, 3, 4	mp2an 430	. . 3 |- { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } e. _V
6		raleq 2749	. . . 4 |- ( x = { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } -> ( A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u ) <-> A. z e. { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u ) ) )
7	6	exbidv 1878	. . 3 |- ( x = { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } -> ( E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u ) <-> E. y A. z e. { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u ) ) )
8		acexmidlemv.choice	. . 3 |- E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u )
9	5, 7, 8	vtocl 2877	. 2 |- E. y A. z e. { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u )
10		eqeq1 2245	. . . . . 6 |- ( s = t -> ( s = (/) <-> t = (/) ) )
11	10	orbi1d 803	. . . . 5 |- ( s = t -> ( ( s = (/) \/ ph ) <-> ( t = (/) \/ ph ) ) )
12	11	cbvrabv 2820	. . . 4 |- { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } = { t e. { (/) , { (/) } } | ( t = (/) \/ ph ) }
13		eqeq1 2245	. . . . . 6 |- ( s = t -> ( s = { (/) } <-> t = { (/) } ) )
14	13	orbi1d 803	. . . . 5 |- ( s = t -> ( ( s = { (/) } \/ ph ) <-> ( t = { (/) } \/ ph ) ) )
15	14	cbvrabv 2820	. . . 4 |- { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } = { t e. { (/) , { (/) } } | ( t = { (/) } \/ ph ) }
16		eqid 2238	. . . 4 |- { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } = { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } }
17	12, 15, 16	acexmidlem2 6072	. . 3 |- ( A. z e. { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u ) -> ( ph \/ -. ph ) )
18	17	exlimiv 1651	. 2 |- ( E. y A. z e. { { s e. { (/) , { (/) } } | ( s = (/) \/ ph ) } , { s e. { (/) , { (/) } } | ( s = { (/) } \/ ph ) } } A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u ) -> ( ph \/ -. ph ) )
19	9, 18	ax-mp 5	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   E!wreu 2530   {crab 2532   _Vcvv 2821   (/)c0 3520   {csn 3705   {cpr 3706   Oncon0 4503

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-iota 5332  df-riota 6028

This theorem is referenced by:  acexmid  6074