Theorem bj-findes 10776

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 10774 for explanations. From this version, it is easy to prove findes 4344. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-findes	|- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Proof of Theorem bj-findes

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . 4 |- F/ y ph
2		nfv 1461	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1504	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1856	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 2833	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1504	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1694	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4157	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2820	. . . 4 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
10	7, 9	imbi12d 232	. . 3 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
11	3, 6, 10	cbvral 2573	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		nfsbc1v 2833	. . 3 |- F/ x [. (/) / x ]. ph
13		sbceq1a 2824	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
14	13	biimprd 156	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
15		sbequ1 1691	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
16		sbceq1a 2824	. . . 4 |- ( x = suc y -> ( ph <-> [. suc y / x ]. ph ) )
17	16	biimprd 156	. . 3 |- ( x = suc y -> ( [. suc y / x ]. ph -> ph ) )
18	12, 4, 5, 14, 15, 17	bj-findis 10774	. 2 |- ( ( [. (/) / x ]. ph /\ A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) ) -> A. x e. om ph )
19	11, 18	sylan2b 281	1 |- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   [wsb 1685   A.wral 2348   [.wsbc 2815   (/)c0 3251   suc csuc 4120   omcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-bd0 10604  ax-bdim 10605  ax-bdan 10606  ax-bdor 10607  ax-bdn 10608  ax-bdal 10609  ax-bdex 10610  ax-bdeq 10611  ax-bdel 10612  ax-bdsb 10613  ax-bdsep 10675  ax-infvn 10736

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332  df-bdc 10632  df-bj-ind 10722

This theorem is referenced by: (None)