Theorem bj-findes 15921

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15919 for explanations. From this version, it is easy to prove findes 4651. (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 1551	. . . 4 |- F/ y ph
2		nfv 1551	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1595	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1967	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 3017	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1595	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1794	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4449	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 3003	. . . 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 2734	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		nfsbc1v 3017	. . 3 |- F/ x [. (/) / x ]. ph
13		sbceq1a 3008	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
14	13	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
15		sbequ1 1791	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
16		sbceq1a 3008	. . . 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 15919	. 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 1373   [wsb 1785   A.wral 2484   [.wsbc 2998   (/)c0 3460   suc csuc 4412   omcom 4638

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4170  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-bd0 15753  ax-bdim 15754  ax-bdan 15755  ax-bdor 15756  ax-bdn 15757  ax-bdal 15758  ax-bdex 15759  ax-bdeq 15760  ax-bdel 15761  ax-bdsb 15762  ax-bdsep 15824  ax-infvn 15881

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639  df-bdc 15781  df-bj-ind 15867

This theorem is referenced by: (None)