Theorem axccd 39429

Description: An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
axccd.1	|- ( ph -> A ~~ om )
axccd.2	|- ( ( ph /\ x e. A ) -> x =/= (/) )

Assertion

Ref	Expression
axccd	|- ( ph -> E. f A. x e. A ( f ` x ) e. x )

Distinct variable groups:   A, f, x   ph, f, x

Proof of Theorem axccd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axccd.1	. . 3 |- ( ph -> A ~~ om )
2		encv 7963	. . . . . 6 |- ( A ~~ om -> ( A e. _V /\ om e. _V ) )
3	2	simpld 475	. . . . 5 |- ( A ~~ om -> A e. _V )
4	1, 3	syl 17	. . . 4 |- ( ph -> A e. _V )
5		breq1 4656	. . . . . 6 |- ( y = A -> ( y ~~ om <-> A ~~ om ) )
6		raleq 3138	. . . . . . 7 |- ( y = A -> ( A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) <-> A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) )
7	6	exbidv 1850	. . . . . 6 |- ( y = A -> ( E. f A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) <-> E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) )
8	5, 7	imbi12d 334	. . . . 5 |- ( y = A -> ( ( y ~~ om -> E. f A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) ) <-> ( A ~~ om -> E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) ) )
9		ax-cc 9257	. . . . 5 |- ( y ~~ om -> E. f A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) )
10	8, 9	vtoclg 3266	. . . 4 |- ( A e. _V -> ( A ~~ om -> E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) )
11	4, 10	syl 17	. . 3 |- ( ph -> ( A ~~ om -> E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) )
12	1, 11	mpd 15	. 2 |- ( ph -> E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) )
13		nfv 1843	. . . . . 6 |- F/ x ph
14		nfra1 2941	. . . . . 6 |- F/ x A. x e. A ( x =/= (/) -> ( f ` x ) e. x )
15	13, 14	nfan 1828	. . . . 5 |- F/ x ( ph /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) )
16		axccd.2	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> x =/= (/) )
17	16	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) /\ x e. A ) -> x =/= (/) )
18		rspa 2930	. . . . . . . 8 |- ( ( A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) /\ x e. A ) -> ( x =/= (/) -> ( f ` x ) e. x ) )
19	18	adantll 750	. . . . . . 7 |- ( ( ( ph /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) /\ x e. A ) -> ( x =/= (/) -> ( f ` x ) e. x ) )
20	17, 19	mpd 15	. . . . . 6 |- ( ( ( ph /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) /\ x e. A ) -> ( f ` x ) e. x )
21	20	ex 450	. . . . 5 |- ( ( ph /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) -> ( x e. A -> ( f ` x ) e. x ) )
22	15, 21	ralrimi 2957	. . . 4 |- ( ( ph /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) -> A. x e. A ( f ` x ) e. x )
23	22	ex 450	. . 3 |- ( ph -> ( A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) -> A. x e. A ( f ` x ) e. x ) )
24	23	eximdv 1846	. 2 |- ( ph -> ( E. f A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) -> E. f A. x e. A ( f ` x ) e. x ) )
25	12, 24	mpd 15	1 |- ( ph -> E. f A. x e. A ( f ` x ) e. x )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   ` cfv 5888   omcom 7065   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cc 9257

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-en 7956

This theorem is referenced by:  axccd2  39430