Theorem bj-findes 15130

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15128 for explanations. From this version, it is easy to prove findes 4617. (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 1539	. . . 4 |- F/ y ph
2		nfv 1539	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1583	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1951	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 2996	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1583	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1782	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4417	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2982	. . . 4 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
10	7, 9	imbi12d 234	. . 3 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
11	3, 6, 10	cbvral 2714	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		nfsbc1v 2996	. . 3 |- F/ x [. (/) / x ]. ph
13		sbceq1a 2987	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
14	13	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
15		sbequ1 1779	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
16		sbceq1a 2987	. . . 4 |- ( x = suc y -> ( ph <-> [. suc y / x ]. ph ) )
17	16	biimprd 158	. . 3 |- ( x = suc y -> ( [. suc y / x ]. ph -> ph ) )
18	12, 4, 5, 14, 15, 17	bj-findis 15128	. 2 |- ( ( [. (/) / x ]. ph /\ A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) ) -> A. x e. om ph )
19	11, 18	sylan2b 287	1 |- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   [wsb 1773   A.wral 2468   [.wsbc 2977   (/)c0 3437   suc csuc 4380   omcom 4604

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-bd0 14962  ax-bdim 14963  ax-bdan 14964  ax-bdor 14965  ax-bdn 14966  ax-bdal 14967  ax-bdex 14968  ax-bdeq 14969  ax-bdel 14970  ax-bdsb 14971  ax-bdsep 15033  ax-infvn 15090

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4386  df-iom 4605  df-bdc 14990  df-bj-ind 15076

This theorem is referenced by: (None)