Theorem bj-findes 16576

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 16574 for explanations. From this version, it is easy to prove findes 4701. (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 1576	. . . 4 |- F/ y ph
2		nfv 1576	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1620	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1992	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 3050	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1620	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1819	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4499	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 3036	. . . 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 2763	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		nfsbc1v 3050	. . 3 |- F/ x [. (/) / x ]. ph
13		sbceq1a 3041	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
14	13	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
15		sbequ1 1816	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
16		sbceq1a 3041	. . . 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 16574	. 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 1397   [wsb 1810   A.wral 2510   [.wsbc 3031   (/)c0 3494   suc csuc 4462   omcom 4688

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4215  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-bd0 16408  ax-bdim 16409  ax-bdan 16410  ax-bdor 16411  ax-bdn 16412  ax-bdal 16413  ax-bdex 16414  ax-bdeq 16415  ax-bdel 16416  ax-bdsb 16417  ax-bdsep 16479  ax-infvn 16536

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929  df-suc 4468  df-iom 4689  df-bdc 16436  df-bj-ind 16522

This theorem is referenced by: (None)