Theorem bj-findes 15711

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15709 for explanations. From this version, it is easy to prove findes 4640. (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 1542	. . . 4 |- F/ y ph
2		nfv 1542	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1586	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1958	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 3008	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1586	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1785	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4438	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2994	. . . 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 2725	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		nfsbc1v 3008	. . 3 |- F/ x [. (/) / x ]. ph
13		sbceq1a 2999	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
14	13	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
15		sbequ1 1782	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
16		sbceq1a 2999	. . . 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 15709	. 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 1776   A.wral 2475   [.wsbc 2989   (/)c0 3451   suc csuc 4401   omcom 4627

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4160  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-bd0 15543  ax-bdim 15544  ax-bdan 15545  ax-bdor 15546  ax-bdn 15547  ax-bdal 15548  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552  ax-bdsep 15614  ax-infvn 15671

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-suc 4407  df-iom 4628  df-bdc 15571  df-bj-ind 15657

This theorem is referenced by: (None)