Theorem bj-findes 14818

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 14816 for explanations. From this version, it is easy to prove findes 4604. (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 1528	. . . 4 |- F/ y ph
2		nfv 1528	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1572	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1939	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 2983	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1572	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1771	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4404	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2969	. . . 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 2701	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		nfsbc1v 2983	. . 3 |- F/ x [. (/) / x ]. ph
13		sbceq1a 2974	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
14	13	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
15		sbequ1 1768	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
16		sbceq1a 2974	. . . 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 14816	. 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 1353   [wsb 1762   A.wral 2455   [.wsbc 2964   (/)c0 3424   suc csuc 4367   omcom 4591

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4131  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-bd0 14650  ax-bdim 14651  ax-bdan 14652  ax-bdor 14653  ax-bdn 14654  ax-bdal 14655  ax-bdex 14656  ax-bdeq 14657  ax-bdel 14658  ax-bdsb 14659  ax-bdsep 14721  ax-infvn 14778

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592  df-bdc 14678  df-bj-ind 14764

This theorem is referenced by: (None)