Theorem bj-findes 13179

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13177 for explanations. From this version, it is easy to prove findes 4517. (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 1508	. . . 4 |- F/ y ph
2		nfv 1508	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1551	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1912	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 2927	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1551	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1744	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4324	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2914	. . . 4 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
10	7, 9	imbi12d 233	. . 3 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
11	3, 6, 10	cbvral 2650	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		nfsbc1v 2927	. . 3 |- F/ x [. (/) / x ]. ph
13		sbceq1a 2918	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
14	13	biimprd 157	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
15		sbequ1 1741	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
16		sbceq1a 2918	. . . 4 |- ( x = suc y -> ( ph <-> [. suc y / x ]. ph ) )
17	16	biimprd 157	. . 3 |- ( x = suc y -> ( [. suc y / x ]. ph -> ph ) )
18	12, 4, 5, 14, 15, 17	bj-findis 13177	. 2 |- ( ( [. (/) / x ]. ph /\ A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) ) -> A. x e. om ph )
19	11, 18	sylan2b 285	1 |- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   [wsb 1735   A.wral 2416   [.wsbc 2909   (/)c0 3363   suc csuc 4287   omcom 4504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-bd0 13011  ax-bdim 13012  ax-bdan 13013  ax-bdor 13014  ax-bdn 13015  ax-bdal 13016  ax-bdex 13017  ax-bdeq 13018  ax-bdel 13019  ax-bdsb 13020  ax-bdsep 13082  ax-infvn 13139

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13039  df-bj-ind 13125

This theorem is referenced by: (None)