Theorem bj-bdfindes 16889

Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 16887 for explanations. From this version, it is easy to prove the bounded version of findes 4745. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-bdfindes.bd	|- BOUNDED ph

Assertion

Ref	Expression
bj-bdfindes	|- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Proof of Theorem bj-bdfindes

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1581	. . . 4 |- F/ y ph
2		nfv 1581	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1625	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1999	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 3070	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1625	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1824	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4542	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 3056	. . . 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 2782	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		bj-bdfindes.bd	. . 3 |- BOUNDED ph
13		nfsbc1v 3070	. . 3 |- F/ x [. (/) / x ]. ph
14		sbceq1a 3061	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
15	14	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
16		sbequ1 1821	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
17		sbceq1a 3061	. . . 4 |- ( x = suc y -> ( ph <-> [. suc y / x ]. ph ) )
18	17	biimprd 158	. . 3 |- ( x = suc y -> ( [. suc y / x ]. ph -> ph ) )
19	12, 13, 4, 5, 15, 16, 18	bj-bdfindis 16887	. 2 |- ( ( [. (/) / x ]. ph /\ A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) ) -> A. x e. om ph )
20	11, 19	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 1402   [wsb 1815   A.wral 2528   [.wsbc 3051   (/)c0 3520   suc csuc 4505   omcom 4732  BOUNDED wbd 16752

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-nul 4254  ax-pr 4341  ax-un 4573  ax-bd0 16753  ax-bdor 16756  ax-bdex 16759  ax-bdeq 16760  ax-bdel 16761  ax-bdsb 16762  ax-bdsep 16824  ax-infvn 16881

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733  df-bdc 16781  df-bj-ind 16867

This theorem is referenced by: (None)